doc/userguide/Makefile.am

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-	ch-intro.tex ch-inst.tex ch-solbase.tex ch-soladv.tex ch-estbase.tex \
-	ch-estadv.tex ch-solbeh.tex ch-estbeh.tex ch-ramsey.tex ch-trouble.tex \
-	P_DynareStruct2.pdf P_flowest.pdf P_MH2.pdf P_ModStruct2.pdf P_ModStruct3.pdf \
-	P_ModStruct4.pdf P_ModStruct5.pdf P_SchorfMod.pdf P_ShockModel2.pdf
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doc/userguide/UserGuide.tex

-\documentclass[a4paper,11pt]{memoir}
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-\newtheorem{definition}{Definition} [chapter]
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-\newtheorem{property}{Property} [chapter]
-\newtheorem{theorem}{Theorem} [chapter]
-\newtheorem{remark}{Remark} [chapter]
-
-%\makeindex
-
-\begin{document}
-\frontmatter 
-
-\includepdf{Graphics/DynareTitle.pdf}
-
-\title{Dynare v4 - User Guide \\Public beta version}
-\author{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\  Tommaso Mancini Griffoli\\ tommaso.mancini@stanfordalumni.org}
-\date{This draft: January 2013}
-
-\maketitle
-
-\thispagestyle{empty}
-
-\newpage
-
-~\vfill
-
-Copyright  2007-2013 Tommaso Mancini Griffoli
-
-\bigskip
-
-Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
-
-\bigskip
-
-A copy of the license can be found at: \url{http://www.gnu.org/licenses/fdl.txt}
-
-\vfill
-
-\newpage
-
-\tableofcontents
-\listoffigures
-
-\chapter*{Work in Progress!}
-This is the second version of the Dynare User Guide which is still work in progress! This means two things. First, please read this with a critical eye and send me comments! Are some areas unclear? Is anything plain wrong? Are some sections too wordy, are there enough examples, are these clear? On the contrary, are there certain parts that just click particularly well? How can others be improved? I'm very interested to get your feedback. \\
-
-The second thing that a work in progress manuscript comes with is a few internal notes. These are mostly placeholders for future work, notes to myself or others of the Dynare development team, or at times notes to you - our readers - to highlight a feature not yet fully stable. Any such notes are marked with two stars (**). \\
-
-Thanks very much for your patience and good ideas. Please write either direclty to myself: tommaso.mancini@stanfordalumni.org, or \textbf{preferably on the Dynare Documentation Forum} available in the \href{http://www.dynare.org/phpBB3}{Dynare Forums}. 
-
-\chapter*{Contacts and Credits} \label{ch:contacts}
-Dynare was originally developed by Michel Juillard in Paris, France. Currently, the development team of Dynare is composed of
-
-\begin{itemize}
-\item Stphane Adjemian \texttt{<stephane.adjemian@univ-lemans.fr>}
-\item Houtan Bastani \texttt{<houtan@dynare.org>}
-\item Michel Juillard \texttt{<michel.juillard@mjui.fr>}
-\item Frdric Karam \texttt{<frederic.karame@univ-lemans.fr>}
-\item Junior Maih \texttt{<junior.maih@gmail.com>}
-\item Ferhat Mihoubi \texttt{<ferhat.mihoubi@cepremap.org>}
-\item George Perendia \texttt{<george@perendia.orangehome.co.uk>}
-\item Johannes Pfeifer \texttt{<jpfeifer@gmx.de>}
-\item Marco Ratto \texttt{<marco.ratto@jrc.ec.europa.eu>}
-\item Sbastien Villemot \texttt{<sebastien@dynare.org>}
-\end{itemize}
-
-Several parts of Dynare use or have strongly benefited from publicly available programs by G. Anderson, F. Collard, L. Ingber, O. Kamenik, P. Klein, S. Sakata, F. Schorfheide, C. Sims, P. Soederlind and R. Wouters.
-
-Finally, the development of Dynare could not have come such a long ways withough an active community of users who continually pose questions, report bugs and suggest new features. The help of this community is gratefully acknowledged.\\
-
-The email addresses above are provided in case you wish to contact any one of the authors of Dynare directly. We nonetheless encourage you to first use the \href{http://www.dynare.org/phpBB3}{Dynare forums} to ask your questions so that other users can benefit from them as well; remember, almost no question is specific enough to interest just one person, and yours is not the exception! 
-
-\mainmatter
-
-\include{ch-intro}
-\include{ch-inst}
-\include{ch-solbase}
-\include{ch-soladv}
-\include{ch-estbase}
-\include{ch-estadv}
-\include{ch-solbeh}
-\include{ch-estbeh}
-\include{ch-ramsey}
-\include{ch-trouble}
-
-\backmatter
-
-\bibliography{DynareBib}
-\bibliographystyle{elsarticle-harv}
-%\printindex
-
-\end{document}
-

doc/userguide/appendix.tex

-\chapter{Appendix}
-This is an example of an appendix!  
\ No newline at end of file

doc/userguide/ch-estadv.tex

-\chapter{Estimating DSGE models - advanced topics} \label{ch:estadv}
-
-This chapter focusses on advanced topics and features of Dynare in the area of model estimation. The chapter begins by presenting a more complex example than the one used for illustration purposes in chapter \ref{ch:estbase}. The goal is to show how Dynare would be used in the more ``realistic'' setting of reproducing a recent academic paper. The chapter then follows with sections on comparing models to one another, and then to BVARs, and ends with a table summarizing where output series are stored and how these can be retrieved. \\
-
-
-\section{Alternative and non-stationary example}
-The example provided in chapter \ref{ch:estbase} is really only useful for illustration purposes. So we thought you would enjoy (and continue learning from!) a more realistic example which reproduces the work in a recent - and highly regarded - academic paper. The example shows how to use Dynare in a more realistic setting, while emphasizing techniques to deal with non-stationary observations and stochastic trends in dynamics. \\
-
-\subsection{Introducing the example}
-The example is drawn from \citet{Schorfheide2000}. This first section introduces the model, its basic intuitions and equations. We will then see in subsequent sections how to estimate it using Dynare. Note that the original paper by Schorfheide mainly focusses on estimation methodologies, difficulties and solutions, with a special interest in model comparison, while the mathematics and economic intuitions of the model it evaluates are drawn from \citet{CogleyNason1994}. That paper should serve as a helpful reference if anything is left unclear in the description below.\\
-
-In essence, the model studied by \citet{Schorfheide2000} is one of cash in advance (CIA). The goal of the paper is to estimate the model using Bayesian techniques, while observing only output and inflation. In the model, there are several markets and actors to keep track of. So to clarify things, figure \ref{fig:schorfmod} sketches the main dynamics of the model. You may want to refer back to the figure as you read through the following sections. 
-\begin{figure} \label{fig:schorfmod}
-\begin{center} 
-\includegraphics[width=1.0\textwidth]{P_SchorfMod} 
-\end{center} 
-\caption[CIA model illustration]{Continuous lines show the circulation of nominal funds, while dashed lines show the flow of real variables.} 
-\end{figure}\\
-
-The economy is made up of three central agents and one secondary agent: households, firms and banks (representing the financial sector), and a monetary authority which plays a minor role. Households maximize their utility function which depends on consumption, $C_t$, and hours worked, $H_t$, while deciding how much money to hold next period in cash, $M_{t+1}$ and how much to deposit at the bank, $D_t$, in order to earn $R_{H,t}-1$ interest. Households therefore solve the problem
-\begin{eqnarray*}
-\substack{\max \\ \{C_t,H_t,M_{t+1},D_t\}} & \mathbb{E}_0 \left[ \sum_{t=0}^\infty \beta^t \left[ (1-\phi) \ln C_t + \phi \ln (1-H_t) \right] \right] \\
-\textrm{s.t.} & P_t C_t \leq M_t - D_t + W_t H_t \\
-& 0 \leq D_t \\
-& M_{t+1} = (M_t - D_t + W_tH_t - P_tC_t) + R_{H,t}D_t + F_t + B_t
-\end{eqnarray*}
-where the second equation spells out the cash in advance constraint including wage revenues, the third the inability to borrow from the bank and the fourth the intertemporal budget constraint emphasizing that households accumulate the money that remains after bank deposits and purchases on goods are deducted from total inflows made up of the money they receive from last period's cash balances, wages, interests, as well as dividends from firms, $F_t$, and from banks, $B_t$, which in both cases are made up of net cash inflows defined below. \\
-
-Banks, on their end, receive cash deposits from households and a cash injection, $X_t$ from the central bank (which equals the net change in nominal money balances, $M_{t+1}-M_t$). It uses these funds to disburse loans to firms, $L_t$, on which they make a net return of $R_{F,t}-1$. Of course, banks are constrained in their loans by a credit market equilibrium condition $L_t \leq X_t + D_t$. Finally, bank dividends, $B_t$ are simply equal to $D_t + R_{F,t}L_t - R_{H,t}D_t - L_t + X_t$. \\
-
-Finally, firms maximize the net present value of future dividends (discounted by the marginal utility of consumption, since they are owned by households) by choosing dividends, next period's capital stock, $K_{t+1}$, labor demand, $N_t$, and loans. Its problem is summarized by
-\begin{eqnarray*}
-\substack{\max \\ \{F_t,K_{t+1},N_{t},L_t\}} & \mathbb{E}_0 \left[ \sum_{t=0}^\infty \beta^{t+1} \frac{F_t}{C_{t+1}P_{t+1}} \right] \\
-\textrm{s.t.} & F_t \leq L_t + P_t \left[ K_t^\alpha (A_t N_t)^{1-\alpha} - K_{t+1} + (1-\delta)K_t \right] - W_tN_t-L_tR_{F,t} \\
-& W_tN_t \leq L_t\\
-\end{eqnarray*}
-where the second equation makes use of the production function \mbox{$Y_t = K_t^\alpha (A_t N_t)^{1-\alpha}$} and the real aggregate accounting constraint (goods market equilibrium) \mbox{$C_t + I_t = Y_t$}, where $I_t=K_{t+1} - (1-\delta)K_t$, and where $\delta$ is the rate of depreciation. Note that it is the firms that engage in investment in this model, by trading off investment for dividends to consumers. The third equation simply specifies that bank loans are used to pay for wage costs. \\
-
-To close the model, we add the usual labor and money market equilibrium equations, $H_t= N_t$ and $P_tC_t=M_t + X_t$, as well as the condition that $R_{H,t}=R_{F,t}$ due to the equal risk profiles of the loans.\\
-
-More importantly, we add a stochastic elements to the model. The model allows for two sources of perturbations, one real, affecting technology and one nominal, affecting the money stock. These important equations are
-\[
-\ln A_t = \gamma + \ln A_{t-1} + \epsilon_{A,t}, \qquad \epsilon_{A,t} \thicksim N(0,\sigma_A^2)
-\]
-and
-
-\[
-\ln m_t = (1-\rho)\ln m^* + \rho \ln m_{t-1} + \epsilon_{M,t}, \qquad \epsilon_{M,t} \thicksim N(0,\sigma_M^2)
-\]
-where $m_t \equiv M_{T+1}/M_t$ is the growth rate of the money stock. Note that theses expressions for trends are not written in the most straightforward manner nor very consistently. But we reproduced them never-the-less to make it easier to compare this example to the original paper. \\
-
-The first equation is therefore a unit root with drift in the log of technology, and the second an autoregressive stationary process in the growth rate of money, but an AR(2) with a unit root in the log of the level of money. This can be seen from the definition of $m_t$ which can be rewritten as $\ln M_{t+1} = \ln M_t + \ln m_t$.\footnote{Alternatively, we could have written the AR(2) process in state space form and realized that the system has an eigenvalue of one. Otherwise said, one is a root of the second order autoregressive lag polynomial. As usual, if the logs of a variable are specified to follow a unit root process, the rate of growth of the series is a stationary stochastic process; see \citet{Hamilton1994}, chapter 15, for details.}\\
-
-When the above functions are maximized, we obtain the following set of first order and equilibrium conditions. We will not dwell on the derivations here, to save space, but encourage you to browse \citet{CogleyNason1994} for additional details. We nonetheless give a brief intuitive explanation of each equation. The system comes down to
-\begin{eqnarray*}
-\mathbb{E}_{t}\bigg\{-\widehat{P}_{t}/\left[ \widehat{C}_{t+1}\widehat{P}_{t+1}m_{t}%
-\right]\bigg\}&=&\beta e^{-\alpha (\gamma +\epsilon_{A,t+1})}P_{t+1}\Big[\alpha\widehat{K}_{t}^{\alpha -1}N_{t+1}^{1-\alpha }+
-(1-\delta )\Big]\\
-&&/\left[ 
-\widehat{c}_{t+2}\widehat{P}_{t+2}m_{t+1}\right] \bigg\}\\
-\widehat{W}_{t}&=&\widehat{L}_{t}/N_{t}\\
-\frac{\phi }{1-\phi }\left[ \widehat{C}_{t}\widehat{P}_{t}/\left(
-1-N_{t}\right) \right]&=&\widehat{L}_{t}/N_{t}\\
-R_{t}&=&(1-\alpha )\widehat{P}_{t}e^{-\alpha (\gamma +\epsilon_{A,t+1})}\widehat{K}_{t-1}^{\alpha}N_{t}^{-\alpha }/\widehat{W}_{t}\\
-\left[ \widehat{C}_{t}\widehat{P}_{t}\right] ^{-1}&=&\beta \left[ \left(
-1-\alpha \right) \widehat{P}_{t}e^{-\alpha (\gamma +\epsilon_{A,t+1})}\widehat{K}_{t-1}^{\alpha}N_{t}^{1-\alpha }
-\right]\\
-&& \times  \mathbb{E}_{t}\left[\widehat{L}_{t}m_{t}\widehat{C}_{t+1}\widehat{P}_{t+1}\right] ^{-1}\\
-\widehat{C}_t+\widehat{K}_t &=& e^{-\alpha(\gamma+\epsilon_{A,t})}\widehat{K}_{t-1}^\alpha N^{1-\alpha}+(1-\delta)e^{-(\gamma+\epsilon_{A,t})}\widehat{K}_{t-1}\\
-\widehat{P}_t\widehat{C} &=& m_t\\
-m_t-1+\widehat{D}_t &=& \widehat{L}_t\\
-\widehat{Y}_t &=& \widehat{K}_{t-1}^\alpha N^{1-\alpha}e^{-\alpha (\gamma+\epsilon_{A,t})}\\
-\ln(m_t) &=& (1-\rho)\ln(m^\star) + \rho\ln(m_{t-1})+\epsilon_{M,t}\\
-\frac{A_t}{A_{t-1}}  \equiv  dA_t & = & \exp( \gamma + \epsilon_{A,t})\\
-Y_t/Y_{t-1} &=& e^{\gamma+\epsilon_{A,t}}\widehat{Y}_t/\widehat{Y}_{t-1}\\
-P_t/P_{t-1} &=& (\widehat{P}_t/\widehat{P}_{t-1})(m_{t-1}/e^{\gamma+\epsilon_{A,t}})
-\end{eqnarray*}
-where, importantly, hats over variables no longer mean deviations from steady state, but instead represent variables that have been made stationary. We come back to this important topic in details in section \ref{sec:nonstat} below. For now, we pause a moment to give some intuition for the above equations. In order, these equations correspond to:
-\begin{enumerate}
-\item The Euler equation in the goods market, representing the tradeoff to the economy of moving consumption goods across time.
-\item The firms' borrowing constraint, also affecting labor demand, as firms use borrowed funds to pay for labor input.
-\item The intertemporal labor market optimality condition, linking labor supply, labor demand, and the marginal rate of substitution between consumption and leisure. 
-\item The equilibrium interest rate in which the marginal revenue product of labor equals the cost of borrowing to pay for that additional unit of labor.
-\item The Euler equation in the credit market, which ensures that giving up one unit of consumption today for additional savings equals the net present value of future consumption.
-\item The aggregate resource constraint.
-\item The money market equilibrium condition equating nominal consumption demand to money demand to money supply to current nominal balances plus money injection. 
-\item The credit market equilibrium condition.
-\item The production function.
-\item The stochastic process for money growth.
-\item The stochastic process for technology.
-\item The relationship between observable variables and stationary variables; more details on these last two equations appear in the following section. 
-\end{enumerate}
-
-\subsection{Declaring variables and parameters}
-This block of the .mod file follows the usual conventions and would look like:\\
-\\
-\texttt{var m P c e W R k d n l Y\_obs P\_obs y dA; \\
-varexo e\_a e\_m;\\
-\\
-parameters alp, bet, gam, mst, rho, psi, del;}\\
-\\
-where the choice of upper and lower case letters is not significant, the first set of endogenous variables, up to $l$, are as specified in the model setup above, and where the last five variables are defined and explained in more details in the section below on declaring the model in Dynare. The exogenous variables are as expected and concern the shocks to the evolution of technology and money balances. \\
-
-\subsection{The origin of non-stationarity} \label{sec:nonstat}
-The problem of non-stationarity comes from having stochastic trends in technology and money. The non-stationarity comes out clearly when attempting to solve the model for a steady state and realizing it does not have one. It can be shown that when shocks are null, real variables grow with $A_t$ (except for labor, $N_t$, which is stationary as there is no population growth), nominal variables grow with $M_t$ and prices with $M_t/A_t$. \textbf{Detrending} therefore involves the following operations (where hats over variables represent stationary variables): for real variables, $\hat q_t=q_t/A_t$, where $q_t = [y_t, c_t, i_t, k_{t+1} ]$. For nominal variables, $\hat{Q}_t = Q_t/M_t$, where $Q_t = [ d_t, l_t, W_t ]$. And for prices, $\hat P_t = P_t \cdot A_t / M_t$. \\
-
-\subsection{Stationarizing variables}
-Let's illustrate this transformation on output, and leave the transformations of the remaining equations as an exercise, if you wish (\citet{CogleyNason1994} includes more details on the transformations of each equation). We stationarize output by dividing its real variables (except for labor) by $A_t$. We define $\widehat{Y}_t$ to equal $Y_t / A_t$ and $\widehat{K}_t$ as $K_t / A_t$. \textsf{\textbf{NOTE!}} Recall from section \ref{sec:modspe} in chapter \ref{ch:solbase}), that in Dynare variables take the time subscript of the period in which they are decided (in the case of the capital stock, today's capital stock is a result of yesterday's decision). Thus, in the output equation, we should actually work with $\widehat{K}_{t-1}=K_{t-1} / A_{t-1}$. The resulting equation made up of stationary variables is
-\begin{eqnarray*}
-\frac{Y_t}{A_t} & = & \left( \frac{K_{t-1}}{A_{t-1}} \right)^\alpha A_t^{1-\alpha} N_t^{1-\alpha} A_t^{-1} A_{t-1}^\alpha \\
-\widehat{Y}_t & = & \widehat{K}_{t-1}^\alpha N_t^{1-\alpha} \left( \frac{A_t}{A_{t-1}} \right)^{-\alpha} \\
-& = & \widehat{K}_{t-1}^\alpha N_t^{1-\alpha} \exp(-\alpha (\gamma + \epsilon_{A,t}))
-\end{eqnarray*}
-where we go from the second to the third line by taking the exponential of both sides of the equation of motion of technology.\\
-
-The above is the equation we retain for the .mod file of Dynare into which we enter:\\
-\\
-\texttt{y=k(-1)\textasciicircum alp*n\textasciicircum (1-alp)*exp(-alp*(gam+e\_a))}\\
-\\
-
-The other equations are entered into the .mod file after transforming them in exactly the same way as the one above. A final transformation to consider, that turns out to be useful since we often deal with the growth rate of technology, is to define \\
-\\
-\texttt{dA = exp(gam+e\_a)}\\
-\\
-by simply taking the exponential of both sides of the stochastic process of technology defined in the model setup above. \\
-
-\subsection{Linking stationary variables to the data}
-And finally, we must make a decision as to our \textbf{non-stationary observations}. We could simply stationarize them by \textbf{working with rates of growth} (which we know are constant). In the case of output, the observable variable would become $Y_t /Y_{t-1}$. We would then have to relate this observable, call it $gy\_{obs}$, to our (stationary) model's variables $\widehat Y_t$ by using the definition that $ \widehat{Y}_t \equiv Y_t/ A_t$. Thus, we add to the model block of the .mod file: \\
-\\
-\texttt{gy\_obs = dA*y/y(-1);}\\
-\\
-where, the $y$ of the .mod file are the stationary $\widehat Y_t$.\\
-
-But, we could also \textbf{work with non-stationary data in levels}. This complicates things somewhat, but illustrates several features of Dynare worth highlighting; we therefore follow this path in the remainder of the example. The result is not very different, though, from what we just saw above. The goal is to add a line to the model block of our .mod file that relates the non stationary observables, call them $Y_{obs}$, to our stationary output, $\widehat Y_t$. We could simply write $Y_{obs} = \widehat Y_t A_t$. But since we don't have an $A_t$ variable, but just a $dA_t$, we we-write the above relationship in ratios. To the .mod file, we therefore add:\\
-\\
-\texttt{Y\_obs/Y\_obs(-1) = dA*y/y(-1);}\\
-\\
-We of course do the same for prices, our other observable variable, except that we use the relationship $P_{obs} = \widehat P_t M_t/A_t$ as noted earlier. The details of the correct transformations for prices are left as an exercise and can be checked against the results below.\\
-
-\subsection{The resulting model block of the .mod file}
-
-\texttt{model;\\
-dA = exp(gam+e\_a);\\
-log(m) = (1-rho)*log(mst) + rho*log(m(-1))+e\_m;\\
--P/(c(+1)*P(+1)*m)+bet*P(+1)*(alp*exp(-alp*(gam+log(e(+1))))*k\textasciicircum (alp-1)\\
-*n(+1)\textasciicircum (1-alp)+(1-del)*exp(-(gam+log(e(+1)))))/(c(+2)*P(+2)*m(+1))=0;\\
-W = l/n;\\
--(psi/(1-psi))*(c*P/(1-n))+l/n = 0;\\
-R = P*(1-alp)*exp(-alp*(gam+e\_a))*k(-1)\textasciicircum alp*n\textasciicircum (-alp)/W;\\
-1/(c*P)-bet*P*(1-alp)*exp(-alp*(gam+e\_a))*k(-1)\textasciicircum alp*n\textasciicircum (1-alp)/\\
-(m*l*c(+1)*P(+1)) = 0;\\
-c+k = exp(-alp*(gam+e\_a))*k(-1)\textasciicircum alp*n\textasciicircum (1-alp)+(1-del)\\
-*exp(-(gam+e\_a))*k(-1);\\
-P*c = m;\\
-m-1+d = l;\\
-e = exp(e\_a);\\
-y = k(-1)\textasciicircum alp*n\textasciicircum (1-alp)*exp(-alp*(gam+e\_a));\\
-Y\_obs/Y\_obs(-1) = dA*y/y(-1);\\
-P\_obs/P\_obs(-1) = (p/p(-1))*m(-1)/dA;\\
-end;}\\
-\\
-where, of course, the input conventions, such as ending lines with semicolons and indicating the timing of variables in parentheses, are the same as those listed in chapter \ref{ch:solbase}. \\
-
-\textsf{\textbf{TIP!}} In the above model block, notice that what we have done is in fact relegated the non-stationarity of the model to just the last two equations, concerning the observables which are, after all, non-stationary. The problem that arises, though, is that we cannot linearize the above system in levels, as the last two equations don't have a steady state. If we first take logs, though, they become linear and it doesn't matter anymore where we calculate their derivative when taking a Taylor expansion of all the equations in the system. Thus, \textbf{when dealing with non-stationary observations, you must log-linearize your model} (and not just linearize it); this is a point to which we will return later. 
-
-\subsection{Declaring observable variables}
-We begin by declaring which of our model's variables are observables. In our .mod file we write\\
-\\
-\texttt{varobs P\_obs Y\_obs;}\\
-\\
-to specify that our observable variables are indeed $P\_obs$ and $Y\_obs$ as noted in the section above. \textsf{\textbf{NOTE!}} Recall from earlier that the number of observed variables must be smaller or equal to the number of shocks such that the model be estimated. If this is not the case, you should add measurement shocks to your model where you deem most appropriate. \\
-
-\subsection{Declaring trends in observable variables}
-
-Recall that we decided to work with the non-stationary observable variables in levels. Both output and prices exhibit stochastic trends. This can be seen explicitly by taking the difference of logs of output and prices to compute growth rates. In the case of output, we make use of the usual (by now!) relationship $Y_t=\widehat Y_t \cdot A_t$. Taking logs of both sides and subtracting the same equation scrolled back one period, we find:
-\[
-\Delta \ln Y_t = \Delta \ln \widehat Y_t + \gamma + \epsilon_{A,t}
-\]
-emphasizing clearly the drift term $\gamma$, whereas we know $\Delta \ln \widehat Y_t$ is stationary in steady state. \\
-
-In the case of prices, we apply the same manipulations to show that:
-\[
-\Delta \ln P_t = \Delta \ln \widehat P_t + \ln m_{t-1} - \gamma - \epsilon_{A,t}
-\]
-
-Note from the original equation of motion of $\ln m_t$ that in steady state, $\ln m_t=\ln m^*$, so that the drift terms in the above equation are $\ln m^* - \gamma$.\footnote{This can also be see from substituting for $\ln m_{t-1}$ in the above equation with the equation of motion of $\ln m_t$ to yield: $\Delta \ln P_t = \Delta \ln \widehat P_t + \ln m^* + \rho(\ln m_{t-2}-\ln m^*) + \epsilon_{M,t} - \gamma - \epsilon_{A,t}$ where all terms on the right hand side are constant, except for $\ln m^*$ and $\gamma$.} \\
-
-In Dynare, any trends, whether deterministic or stochastic (the drift term) must be declared up front. In the case of our example, we therefore write (in a somewhat cumbersome manner)\\
-\\
-\texttt{observation\_trends;\\
-P\_obs (log(mst)-gam);\\
-Y\_obs (gam);\\
-end;}\\
-
-In general, the command \texttt{observation\_trends} specifies linear trends as a function of model parameters for the observed variables in the model.\\
-
-
-\subsection{Specifying the steady state}
-Declaring the steady state is just as explained in details and according to the same syntax explained in chapter \ref{ch:solbase}, covering the \texttt{initval}, \texttt{steady} and \texttt{check} commands. In chapter \ref{ch:estbase}, section \ref{sec:ssest}, we also discussed the usefulness of providing an external Matlab file to solve for the steady state. In this case, you can find the corresponding steady state file in the \textsl{models} folder under \textsl{UserGuide}. The file is called \textsl{fs2000ns\_steadystate.m}. There are some things to notice. First, the output of the function is the endogenous variables at steady state, the \texttt{ys} vector. The \texttt{check=0} limits steady state values to real numbers. Second, notice the declaration of parameters at the beginning; intuitive, but tedious... This functionality may be updated in later versions of Dynare. Third, note that the file is really only a sequential set of equalities, defining each variable in terms of parameters or variables solved in the lines above. So far, nothing has changed with respect to the equivalent file of chapter \ref{ch:estbase}. The only novelty is the declaration of the non-stationary variables, $P\_obs$ and $Y\_obs$ which take the value of 1. This is Dynare convention and must be the case for all your non-stationary variables. 
-
-\subsection{Declaring priors}
-We expand our .mod file with the following information: \\
-\\
-\texttt{estimated\_params;\\
-alp, beta\_pdf, 0.356, 0.02; \\ 
-bet, beta\_pdf, 0.993, 0.002; \\
-gam, normal\_pdf, 0.0085, 0.003; \\
-mst, normal\_pdf, 1.0002, 0.007; \\
-rho, beta\_pdf, 0.129, 0.223;\\
-psi, beta\_pdf, 0.65, 0.05;\\
-del, beta\_pdf, 0.01, 0.005;\\
-stderr e\_a, inv\_gamma\_pdf, 0.035449, inf;\\
-stderr e\_m, inv\_gamma\_pdf, 0.008862, inf;\\
-end;}\\
-\\
-
-\subsection{Launching the estimation}
-We add the following commands to ask Dynare to run a basic estimation of our model:\\
-\\
-\texttt{estimation(datafile=fsdat,nobs=192,loglinear,mh\_replic=2000,\\
-mode\_compute=4,mh\_nblocks=2,mh\_drop=0.45,mh\_jscale=0.65,\\
-diffuse\_filter);}\\
-
-\textsf{\textbf{NOTE!}} Since P\_obs and Y\_obs inherit the unit root characteristics of their driving variables, technology and money, we must tell Dynare to use a diffuse prior (infinite variance) for their initialization in the Kalman filter. Note that for stationary variables, the unconditional covariance matrix of these variables is used for initialization. The algorithm to compute a true diffuse prior is taken from \citet{DurbinKoopman2001}. To give these instructions to Dynare, we use the \texttt{diffuse\_filter} option. Also note that you don't need to use the diffuse filter for any non-stationary model, only for when there is a unit-root, \textit{i.e.} a stochastic trend; you don't need to use a diffuse initial condition in the case of a deterministic trend, since the variance is finite.\\
-
-\textsf{\textbf{NOTE!}} As mentioned earlier, we need to instruct Dynare to log-linearize our model, since it contains non-linear equations in non-stationary variables. A simple linearization would fail as these variables do not have a steady state. Fortunately, taking the log of the equations involving non-stationary variables does the job of linearizing them.\\
-
-\subsection{The complete .mod file}
-We have seen each part of the .mod separately; it's now time to get a picture of what the complete file looks like. For convenience, the file also appears in the \textsl{models} folder under \textsl{UserGuide} in your Dynare installation. The file is called \texttt{fs2000ns.mod}. \\
-\\
-\texttt{var m P c e W R k d n l Y\_obs P\_obs y dA; \\
-varexo e\_a e\_m;\\
-parameters alp, bet, gam, mst, rho, psi, del;\\
-\\
-model;\\
-dA = exp(gam+e\_a);\\
-log(m) = (1-rho)*log(mst) + rho*log(m(-1))+e\_m;\\
--P/(c(+1)*P(+1)*m)+bet*P(+1)*(alp*exp(-alp*(gam+log(e(+1))))\\
-*k\textasciicircum (alp-1)*n(+1)\textasciicircum (1-alp)+(1-del)\\
-*exp(-(gam+log(e(+1)))))/(c(+2)*P(+2)*m(+1))=0;\\
-W = l/n;\\
--(psi/(1-psi))*(c*P/(1-n))+l/n = 0;\\
-R = P*(1-alp)*exp(-alp*(gam+e\_a))*k(-1)\textasciicircum alp*n\textasciicircum (-alp)/W;\\
-1/(c*P)-bet*P*(1-alp)*exp(-alp*(gam+e\_a))*k(-1)\textasciicircum alp*n\textasciicircum (1-alp)/\\(m*l*c(+1)*P(+1)) = 0;\\
-c+k = exp(-alp*(gam+e\_a))*k(-1)\textasciicircum alp*n\textasciicircum (1-alp)+(1-del)\\*exp(-(gam+e\_a))*k(-1);\\
-P*c = m;\\
-m-1+d = l;\\
-e = exp(e\_a);\\
-y = k(-1)\textasciicircum alp*n\textasciicircum (1-alp)*exp(-alp*(gam+e\_a));\\
-Y\_obs/Y\_obs(-1) = dA*y/y(-1);\\
-P\_obs/P\_obs(-1) = (p/p(-1))*m(-1)/dA;\\
-end;\\
-\\
-varobs P\_obs Y\_obs;\\
-\\
-observation\_trends;\\
-P\_obs (log(mst)-gam);\\
-Y\_obs (gam);\\
-end;\\
-\\
-initval;\\
-k = 6;\\
-m = mst;\\
-P = 2.25;\\
-c = 0.45;\\
-e = 1;\\
-W = 4;\\
-R = 1.02;\\
-d = 0.85;\\
-n = 0.19;\\
-l = 0.86;\\
-y = 0.6;\\
-dA = exp(gam);\\
-end;\\
-\\
-// the above is really only useful if you want to do a stoch\_simul\\
-// of your model, since the estimation will use the Matlab\\
-// steady state file also provided and discussed above.\\
-\\
-estimated\_params;\\
-alp, beta\_pdf, 0.356, 0.02; \\ 
-bet, beta\_pdf, 0.993, 0.002; \\
-gam, normal\_pdf, 0.0085, 0.003; \\
-mst, normal\_pdf, 1.0002, 0.007; \\
-rho, beta\_pdf, 0.129, 0.223;\\
-psi, beta\_pdf, 0.65, 0.05;\\
-del, beta\_pdf, 0.01, 0.005;\\
-stderr e\_a, inv\_gamma\_pdf, 0.035449, inf;\\
-stderr e\_m, inv\_gamma\_pdf, 0.008862, inf;\\
-end;\\
-\\
-estimation(datafile=fsdat,nobs=192,loglinear,mh\_replic=2000,\\
-mode\_compute=4,mh\_nblocks=2,mh\_drop=0.45,mh\_jscale=0.65,\\
-diffuse\_filter);}\\
-\\
-
-\subsection{Summing it up}
-The explanations given above of each step necessary to translate the \citet{Schorfheide2000} example into language that Dynare can understand and process was quite lengthy and involved a slew of new commands and information. It may therefore be useful, to gain a ``bird's eyeview'' on what we have just accomplished, and summarize the most important steps at a high level. This is done in figure \ref{fig:estsumm}.\\
-\begin{figure} \label{fig:estsumm}
-\begin{center} 
-\includegraphics[width=1.0\textwidth]{P_flowest} 
-\end{center} 
-\caption[Steps of model estimation]{At a high level, there are five basic steps to translate a model into Dynare for successful estimation.} 
-\end{figure}\\
-
-
-\section{Comparing models based on their posterior distributions}
-** TBD
-
-\section{Where is your output stored?}
-The output from estimation can be extremely varied, depending on the instructions you give Dynare. The \href{http://www.dynare.org/documentation-and-support/manual}{Reference Manual} overviews the complete set of potential output files and describes where you can find each one. 
\ No newline at end of file

doc/userguide/ch-estbase.tex

-\chapter{Estimating DSGE models - basics} \label{ch:estbase}
-
-As in the chapter \ref{ch:solbase}, this chapter is structured around an example. The goal of this chapter is to lead you through the basic functionality in Dynare to estimate models using Bayesian techniques, so that by the end of the chapter you should have the capacity to estimate a model of your own. This chapter is therefore very practically-oriented and abstracts from the underlying computations that Dynare undertakes to estimate a model; that subject is instead covered in some depth in chapter \ref{ch:estbeh}, while more advanced topics of practical appeal are discussed in chapter \ref{ch:estadv}. \\
-
-\section{Introducing an example} \label{sec:modsetup}
-The example introduced in this chapter is particularly basic. This is for two reasons. First, we did not want to introduce yet another example in this section; there's enough new material to keep you busy. Instead, we thought it would be easiest to simply continue working with the example introduced in chapter \ref{ch:solbase} with which you are probably already quite familiar. Second, the goal of the example in this chapter is really to explain features in context, but not necessarily to duplicate a ``real life scenario'' you would face when writing a paper. Once you feel comfortable with the content of this chapter, though, you can always move on to chapter \ref{ch:estadv} where you will find a full-fledged replication of a recent academic paper, featuring a non-stationary model. \\
-
-Recall from chapter \ref{ch:solbase} that we are dealing with an RBC model with monopolistic competition. Suppose we had data on business cycle variations of output. Suppose also that we thought our little RBC model did a good job of reproducing reality. We could then use Bayesian methods to estimate the parameters of the model: $\alpha$, the capital share of output, $\beta$, the discount factor, $\delta$, the depreciation rate, $\psi$, the weight on leisure in the utility function, $\rho$, the degree of persistence in productivity, and $\epsilon$, the markup parameter. Note that in Bayesian estimation, the condition for undertaking estimation is that there be at least as many shocks as there are observables (a less stringent condition than for maximum likelihood estimation). It may be that this does not allow you to identify all your parameters - yielding posterior distributions identical to prior distributions - but the Bayesian estimation procedure would still run successfully. Let's see how to go about doing this. 
-
-\section{Declaring variables and parameters}
-To input the above model into Dynare for estimation purposes, we must first declare the model's variables in the preamble of the .mod file. This is done exactly as described in chapter \ref{ch:solbase} on solving DSGE models. We thus begin the .mod file with:\\
-\\
-\texttt{var y c k i l y\_l w r  z;\\
-varexo e;\\
-\\
-parameters beta psi delta alpha rho epsilon;\\}\\
-\\
-
-\section{Declaring the model}
-Suppose that the equation of motion of technology is a \textbf{stationary} AR(1) with an autoregressive parameter, $\rho$, less than one. The model's variables would therefore be stationary and we can proceed without complications. The alternative scenario with \textbf{non-stationary variables} is more complicated and dealt with in chapter \ref{ch:estadv} (in the additional example). In the stationary case, our model block would look exactly as in chater \ref{ch:solbase}:\\
-\\
-\texttt{model;\\
-  (1/c) = beta*(1/c(+1))*(1+r(+1)-delta);\\
-  psi*c/(1-l) = w;\\
-  c+i = y;\\
-  y = (k(-1)\textasciicircum alpha)*(exp(z)*l)\textasciicircum (1-alpha);\\
-  w = y*((epsilon-1)/epsilon)*(1-alpha)/l;\\
-  r = y*((epsilon-1)/epsilon)*alpha/k(-1);\\
-  i = k-(1-delta)*k(-1);\\
-  y\_l = y/l;\\
-  z = rho*z(-1)+e;\\
-end;}\\
-\\
-
-\section{Declaring observable variables}
-This should not come as a surprise. Dynare must know which variables are observable for the estimation procedure. \textsf{\textbf{NOTE!}} These variables must be available in the data file, as explained in section \ref{sec:estimate} below. For the moment, we write:\\
-\\
-\texttt{varobs y;}\\
-
-\section{Specifying the steady state} \label{sec:ssest}
-Before Dynare estimates a model, it first linearizes it around a steady state. Thus, a steady state must exist for the model and although Dynare can calculate it, we must give it a hand by declaring approximate values for the steady state. This is just as explained in details and according to the same syntax outlined in chapter \ref{ch:solbase}, covering the \texttt{initval}, \texttt{steady} and \texttt{check} commands. In fact, as this chapter uses the same model as that outlined in chapter \ref{ch:solbase}, the steady state block will look exactly the same.\\
-
-\textsf{\textbf{TIP!}} During estimation, in finding the posterior mode, Dynare recalculates the steady state of the model at each iteration of the optimization routine (more on this later), based on the newest round of parameters available. Thus, by providing approximate initial values and relying solely on the built-in Dynare algorithm to find the steady state (a numerical procedure), you will significantly slow down the computation of the posterior mode. Dynare will end up spending 60 to 70\% of the time recalculating steady states. It is much more efficient to write an external \textbf{Matlab steady state file} and let Dynare use that file to find the steady state of your model by algebraic procedure. For more details on writing an external Matlab file to find your model's steady state, please refer to section \ref{sec:findsteady} of chapter \ref{ch:solbase}. \\
-
-
-\section{Declaring priors}
-Priors play an important role in Bayesian estimation and consequently deserve a central role in the specification of the .mod file. Priors, in Bayesian estimation, are declared as a distribution. The general syntax to introduce priors in Dynare is the following:\\
-\\
-\texttt{estimated\_params;\\
-PARAMETER NAME, PRIOR\_SHAPE, PRIOR\_MEAN, PRIOR\_STANDARD\_ERROR [, PRIOR 3$^{\textrm{rd}}$ PARAMETER] [,PRIOR 4$^{\textrm{th}}$ PARAMETER] ; \\
-end; }\\
-\\
-where the following table defines each term more clearly\\
-\\
-    \begin{tabular}{llc}
-    \textbf{\underline{PRIOR\_SHAPE}} & \textbf{\underline{Corresponding distribution}} & \textbf{\underline{Range}} \\
-      NORMAL\_PDF & $N(\mu,\sigma)$ & $\mathbb{R}$\\
-      GAMMA\_PDF &  $G_2(\mu,\sigma,p_3)$ & $[p_3,+\infty)$\\
-        BETA\_PDF & $B(\mu,\sigma,p_3,p_4)$ & $[p_3,p_4]$\\
-        INV\_GAMMA\_PDF & $IG_1(\mu,\sigma)$ & $\mathbb R^+$\\
-UNIFORM\_PDF& $U(p_3,p_4)$ &  $[p_3,p_4]$
-    \end{tabular}\\
-    \\
-    \\
-    where $\mu$ is the \texttt{PRIOR\_MEAN}, $\sigma$ is the \texttt{PRIOR\_STANDARD\_ERROR}, $p_3$ is the \texttt{PRIOR 3$^{\textrm{rd}}$ PARAMETER} (whose default is 0) and $p_4$ is the \texttt{PRIOR 4$^{\textrm{th}}$ PARAMETER} (whose default is 1).  \textsf{\textbf{TIP!}} When specifying a uniform distribution between 0 and 1 as a prior for a parameter, say $\alpha$, you therefore have to put two empty spaces for parameters $\mu$ and $\sigma$, and then specify parameters $p3$ and $p4$, since the uniform distribution only takes $p3$ and $p4$ as arguments. For instance, you would write \texttt{alpha, uniform\_pdf, , , 0,1;} \\
-    
-For a more complete review of all possible options for declaring priors, as well as the syntax to declare priors for maximum likelihood estimation (not Bayesian), see the \href{http://www.dynare.org/documentation-and-support/manual}{Reference Manual}. Note also that if some parameters in a model are calibrated and are not to be estimated, you should declare them as such, by using the \texttt{parameters} command and its related syntax, as explained in chapter \ref{ch:solbase}. \\
-
-\textsf{\textbf{TIP!}} Choosing the appropriate prior for your parameters is a tricky, yet very important endeavor. It is worth spending time on your choice of priors and to test the robustness of your results to your priors. Some considerations may prove helpful. First, think about the domain of your prior over each parameter. Should it be bounded? Should it be opened on either or both sides? Remember also that if you specify a probability of zero over a certain domain in your prior, you will necessarily also find a probability of zero in your posterior distribution. Then, think about the shape of your prior distribution. Should it be symmetric? Skewed? If so, on which side? You may also go one step further and build a distribution for each of your parameters in your mind. Ask yourself, for instance, what is the probability that your parameter is bigger than a certain value, and repeat the exercise by incrementally decreasing that value. You can then pick the standard distribution that best fits your perceived distribution. Finally, instead of describing here the shapes and properties of each standard distribution available in Dynare, you are instead encouraged to visualize these distributions yourself, either in a statistics book or on the Web. \\
-    
-\textsf{\textbf{TIP!}} It may at times be desirable to estimate a transformation of a parameter appearing in the model, rather than the parameter itself. In such a case, it is possible to declare the parameter to be estimated in the parameters statement and to define the transformation at the top of the model section, as a Matlab expression, by adding a pound sign (\#) at the beginning of the corresponding line. For example, you may find it easier to define a prior over the discount factor, $\beta$, than its inverse which often shows up in Euler equations. Thus you would write:\\
-\\
-\texttt{model;\\
-      \# sig = 1/bet;\\
-      c = sig*c(+1)*mpk;\\
-      end;\\
-\\
-      estimated\_params;\\
-      bet,normal\_pdf,1,0.05;\\
-      end;}\\
-
-\textsf{\textbf{TIP!}} Finally, another useful command to use is the \texttt{estimated\_params\_init} command which declares numerical initial values for the optimizer when these are different from the prior mean. This is especially useful when redoing an estimation - if the optimizer got stuck the first time around, or needing a greater number of draws in the Metropolis-Hastings algorithm - and wanting to enter the posterior mode as initial values for the parameters instead of a prior. The \href{http://www.dynare.org/documentation-and-support/manual}{Reference Manual} gives more details as to the exact syntax of this command.\\
-
-Coming back to our basic example, we would write:\\
-\\
-\texttt{estimated\_params;\\
-alpha, beta\_pdf, 0.35, 0.02; \\ 
-beta, beta\_pdf, 0.99, 0.002; \\
-delta, beta\_pdf, 0.025, 0.003;\\
-psi, gamma\_pdf, 1.75, 0.1;\\
-rho, beta\_pdf, 0.95, 0.05;\\
-epsilon, gamma\_pdf, 10, 0.5;\\
-stderr e, inv\_gamma\_pdf, 0.01, inf;\\
-end;}\\
-
-
-\section{Launching the estimation} \label{sec:estimate}
-To ask Dynare to estimate a model, all that is necessary is to add the command \texttt{estimation} at the end of the .mod file. Easy enough. But the real complexity comes from the options available for the command (to be entered in parentheses and sequentially, separated by commas, after the command \texttt{estimation}). Below, we list the most common and useful options, and encourage you to view the \href{http://www.dynare.org/documentation-and-support/manual}{Reference Manual} for a complete list. \\
-\begin{enumerate}
-\item datafile = FILENAME: the datafile (a .m file, a .mat file, or an .xls file). Note that observations do not need to show up in any order, but vectors of observations need to be named with the same names as those in \texttt{var\_obs}. In Excel files, for instance, observations could be ordered in columns, and variable names would show up in the first cell of each column.
-\item nobs = INTEGER: the number of observations to be used (default: all observations in the file) 
-\item first\_obs = INTEGER: the number of the first observation to be used (default = 1). This is useful when running loops, or instance, to divide the observations into sub-periods. 
-\item prefilter = 1: the estimation procedure demeans the data (default=0, no prefiltering). This is useful if model variables are in deviations from steady state, for instance, and therefore have zero mean. Demeaning the observations would also impose a zero mean on the observed variables.  
-\item nograph: no graphs should be plotted.
-\item conf\_sig = \{INTEGER | DOUBLE \}: the level for the confidence intervals reported in the results (default = 0.90) 
-\item mh\_replic = INTEGER: number of replication for Metropolis Hasting algorithm. For the time being, mh\_replic 
-should be larger than 1200 (default = 20000) 
-\item mh\_nblocks = INTEGER: number of parallel chains for Metropolis Hasting algorithm (default = 2). Despite this low default value, it is advisable to work with a higher value, such as 5 or more. This improves the computation of between group variance of the parameter means, one of the key criteria to evaluate the efficiency of the Metropolis-Hastings to evaluate the posterior distribution. More details on this subject appear in Chapter \ref{ch:estadv}.  
-\item mh\_drop = DOUBLE: the fraction of initially generated parameter vectors to be dropped before using posterior 
-simulations (default = 0.5; this means that the first half of the draws from the Metropolis-Hastings are discarded). 
-\item mh\_jscale = DOUBLE: the scale to be used for the jumping distribution in MH algorithm. The default value is 
-rarely satisfactory. This option must be tuned to obtain, ideally, an acceptance rate of 25\% in the Metropolis- Hastings algorithm (default = 0.2). The idea is not to reject or accept too often a candidate parameter; the literature has settled on a value of between 0.2 and 0.4. If the acceptance rate were too high, your Metropolis-Hastings iterations would never visit the tails of a distribution, while if it were too low, the iterations would get stuck in a subspace of the parameter range. Note that the acceptance rate drops if you increase the scale used in the jumping distribution and vice a versa.
-\item mh\_init\_scale=DOUBLE: the scale to be used for drawing the initial value of the Metropolis-Hastings chain (default=2*mh\_jscale). The idea here is to draw initial values from a stretched out distribution in order to maximize the chances of these values not being too close together, which would defeat the purpose of running several blocks of Metropolis-Hastings chains.
-\item mode\_file=FILENAME: name of the file containing previous value for the mode. When computing the mode, 
-Dynare stores the mode (xparam1) and the hessian (hh) in a file called MODEL NAME\_mode. This is a particularly helpful option to speed up the estimation process if you have already undertaken initial estimations and have values of the posterior mode. 
-\item mode\_compute=INTEGER: specifies the optimizer for the mode computation. 
-\subitem 0: the mode isnÕt computed. mode\_file must be specified 
-\subitem 1: uses Matlab fmincon (see the \href{http://www.dynare.org/documentation-and-support/manual}{Reference Manual} to set options for this command).
-\subitem 2: uses Lester IngberÕs Adaptive Simulated Annealing. 
-\subitem 3: uses Matlab fminunc. 
-\subitem 4 (default): uses Chris SimÕs csminwel. 
-\item mode\_check: when mode\_check is set, Dynare plots the minus of the posterior density for values around the computed mode 
-for each estimated parameter in turn. This is helpful to diagnose problems with the optimizer. A clear indication of a problem would be that the mode is not at the trough (bottom of the minus) of the posterior distribution. 
-\item load\_mh\_file: when load\_mh\_file is declared, Dynare adds to previous Metropolis-Hastings simulations instead 
-of starting from scratch. Again, this is a useful option to speed up the process of estimation.
-\item nodiagnostic: doesnÕt compute the convergence diagnostics for Metropolis-Hastings (default: diagnostics are computed and 
-displayed). Actually seeing if the various blocks of Metropolis-Hastings runs converge is a powerful and useful option to build confidence in your model estimation. More details on these diagnostics are given in Chapter \ref{ch:estadv}.
-\item bayesian\_irf: triggers the computation of the posterior distribution of impulse response functions (IRFs). The length of the IRFs are controlled by the irf option, as specified in chapter \ref{ch:solbase} when discussing the options for \texttt{stoch\_simul}. To build the posterior distribution of the IRFs, Dynare pulls parameter and shock values from the corresponding estimated distributions and, for each set of draws, generates an IRF. Repeating this process often enough generates a distribution of IRFs. \textsf{\textbf{TIP!}} If you stop the estimation procedure after calculating the posterior mode, or carry out maximum likelihood estimation, only the corresponding parameter estimates will be used to generate the IRFs. If you instead carry out a full Metropolis-Hastings estimation, on the other hand, the IRFs will use the parameters the posterior distributions, including the variance of the shocks. 
-\item All options available for stoch\_simul can simply be added to the above options, separated by commas. To view a list of these options, either see the \href{http://www.dynare.org/documentation-and-support/manual}{Reference Manual} or section \ref{sec:compute} of chapter \ref{ch:solbase}.
-\item moments\_varendo: triggers the computation of the posterior distribution of the theoretical moments of the endogenous variables as in \texttt{stoch\_simul} (the posterior distribution of the variance decomposition is also included). ** will be implemented shortly - if not already - in Dynare version 4. 
-\item filtered\_vars: triggers the computation of the posterior distribution of filtered endogenous variables and shocks. See the note below on the difference between filtered and smoothed shocks. ** will be implemented shortly - if not already - in Dynare version 4. 
-\item smoother: triggers the computation of the posterior distribution of smoothed endogenous variables and shocks. Smoothed shocks are a reconstruction of the values of unobserved shocks over the sample, using all the information contained in the sample of observations. Filtered shocks, instead, are built only based on knowing past information. To calculate one period ahead prediction errors, for instance, you should use filtered, not smoothed variables.
-\item forecast = INTEGER: computes the posterior distribution of a forecast on INTEGER periods after the end of the sample used in estimation. The corresponding graph includes one confidence interval describing uncertainty due to parameters and one confidence interval describing uncertainty due to parameters and future shocks. Note that Dynare cannot forecast out of the posterior mode. You need to run Metropolis-Hastings iterations before being able to run forecasts on an estimated model. Finally, running a forecast is very similar to an IRF, as in \texttt{bayesian\_irf}, except that the forecast does not begin at a steady state, but simply at the point corresponding to the last set of observations. The goal of undertaking a forecast is to see how the system returns to steady state from this starting point. Of course, as observation do not exist for all variables, those necessary are reconstructed by sampling out of the posterior distribution of parameters. Again, repeating this step often enough yields a posterior distribution of the forecast. ** will be implemented shortly - if not already - in Dynare version 4. 
-\end{enumerate}
-
-\textsf{\textbf{TIP!}} Before launching estimation it is a good idea to make sure that your model is correctly declared, that a steady state exists and that it can be simulated for at least one set of parameter values. You may therefore want to create a test version of your .mod file. In this test file, you would comment out or erase the commands related to estimation, remove the prior estimates for parameter values and replace them with actual parameter values in the preamble, remove any non-stationary variables from your model, add a \texttt{shocks} block, make sure you have \texttt{steady} and possibly \texttt{check} following the \texttt{initval} block if you do not have exact steady state values and run a simulation using \texttt{stoch\_simul} at the end of your .mod file. Details on model solution and simulation can be found in Chapter \ref{ch:solbase}. \\
-
-Finally, coming back to our example, we could choose a standard option:\\
-\\
-\texttt{estimation(datafile=simuldataRBC,nobs=200,first\_obs=500,\\
-mh\_replic=2000,mh\_nblocks=2,mh\_drop=0.45,mh\_jscale=0.8,\\
-mode\_compute=4); }\\
-
-This ends our description of the .mod file.
-
-\section{The complete .mod file}
-To summarize and to get a complete perspective on our work so far, here is the complete .mod file for the estimation of our very basic model. You can find the corresponding file in the \textsl{models} folder under \textsl{UserGuide} in your installation of Dynare. The file is called \textsl{RBC\_Est.mod}.\\
-\\
-\texttt{var y c k i l y\_l w r  z;\\
-varexo e;\\
-parameters beta psi delta alpha rho epsilon;\\
-\\
-model;\\
-  (1/c) = beta*(1/c(+1))*(1+r(+1)-delta);\\
-  psi*c/(1-l) = w;\\
-  c+i = y;\\
-  y = (k(-1)\textasciicircum alpha)*(exp(z)*l)\textasciicircum (1-alpha);\\
-  w = y*((epsilon-1)/epsilon)*(1-alpha)/l;\\
-  r = y*((epsilon-1)/epsilon)*alpha/k(-1);\\
-  i = k-(1-delta)*k(-1);\\
-  y\_l = y/l;\\
-  z = rho*z(-1)+e;\\
-end;\\
-\\
-varobs y;\\
-\\
-initval;\\
-  k = 9;\\
-  c = 0.76;\\
-  l = 0.3;\\
-  w = 2.07;\\
-  r = 0.03;\\
-  z = 0; \\
-  e = 0;\\
-end;\\
-\\
-estimated\_params;\\
-alpha, beta\_pdf, 0.35, 0.02; \\ 
-beta, beta\_pdf, 0.99, 0.002; \\
-delta, beta\_pdf, 0.025, 0.003;\\
-psi, gamma\_pdf, 1.75, 0.1;\\
-rho, beta\_pdf, 0.95, 0.05;\\
-epsilon, gamma\_pdf, 10, 0.5;\\
-stderr e, inv\_gamma\_pdf, 0.01, inf;\\
-end;\\
-\\
-estimation(datafile=simuldataRBC,nobs=200,first\_obs=500,\\
-mh\_replic=2000,mh\_nblocks=2,mh\_drop=0.45,mh\_jscale=0.8,\\
-mode\_compute=4); }
-\\
-
-
-\section{Interpreting output}
-
-As in the case of model solution and simulation, Dynare returns both tabular and graphical output. On the basis of the options entered in the example .mod file above, Dynare will display the following results.\\
-
-\subsection{Tabular results}
-The first results to be displayed (and calculated from a chronological standpoint) are the steady state results. Note the dummy values of 1 for the non-stationary variables Y\_obs and P\_obs. These results are followed by the eigenvalues of the system, presented in the order in which the endogenous variables are declared at the beginning of the .mod file. The table of eigenvalues is completed with a statement about the Blanchard-Kahn condition being met - hopefully!\\
-
-The next set of results are for the numerical iterations necessary to find the posterior mode, as explained in more details in Chapter \ref{ch:estadv}. The improvement from one iteration to the next reaches zero, Dynare give the value of the objective function (the posterior Kernel) at the mode and displays two important table summarizing results from posterior maximization.\\
-
-The first table summarizes results for parameter values. It includes: prior means, posterior mode, standard deviation and t-stat of the mode (based on the assumption of a Normal, probably erroneous when undertaking Bayesian estimation, as opposed to standard maximum likelihood), as well as the prior distribution and standard deviation (pstdev). It is followed by a second table summarizing the same results for the shocks. It may be entirely possible that you get an infinite value for a standard deviation, this is simply the limit case of the inverse gamma distribution.\\
-
-\subsection{Graphical results}
-** corresponding graphs will be reproduced below.\\
-
-The first figure comes up soon after launching Dynare as little computation is necessary to generate it. The figure returns a graphical representation of the priors for each parameter of interest. \\
-
-The second set of figures displays ``MCMC univariate diagnostics'', where MCMC stands for Monte Carlo Markov Chains. This is the main source of feedback to gain confidence, or spot a problem, with results. Recall that Dynare completes several runs of Metropolis-Hastings simulations (as many as determined in the option \texttt{mh\_nblocks}, each time starting from a different initial value). If the results from one chain are sensible, and the optimizer did not get stuck in an odd area of the parameter subspace, two things should happen. First, results within any of the however many iterations of Metropolis-Hastings simulation should be similar. And second, results between the various chains should be close. This is the idea of what the MCMC diagnostics track. \\
-
-More specifically, the red and blue lines on the charts represent specific measures of the parameter vectors both within and between chains. For the results to be sensible, these should be relatively constant (although there will always be some variation) and they should converge. Dynare reports three measures: ``interval'', being constructed from an 80\% confidence interval around the parameter mean, ``m2'', being a measure of the variance and ``m3'' based on third moments. In each case, Dynare reports both the within and the between chains measures. The figure entitled ``multivariate diagnostic'' presents results of the same nature, except that they reflect an aggregate measure based on the eigenvalues of the variance-covariance matrix of each parameter.\\
-
-In our example above, you can tell that indeed, we obtain convergence and relative stability in all measures of the parameter moments. Note that the horizontal axis represents the number of Metropolis-Hastings iterations that have been undertaken, and the vertical axis the measure of the parameter moments, with the first, corresponding to the measure at the initial value of the Metropolis-Hastings iterations.\\
-
-\textsf{\textbf{TIP!}} If the plotted moments are highly unstable or do not converge, you may have a problem of poor priors. It is advisable to redo the estimation with different priors. If you have trouble coming up with a new prior, try starting with a uniform and relatively wide prior and see where the data leads the posterior distribution. Another approach is to undertake a greater number of Metropolis-Hastings simulations.\\
-
-The first to last figure - figure 6 in our example - displays the most interesting set of results, towards which most of the computations undertaken by Dynare are directed: the posterior distribution. In fact, the figure compares the posterior to the prior distribution (black vs. grey lines). In addition, on the posterior distribution, Dynare plots a green line which represents the posterior mode. These allow you to make statements about your data other than simply concerning the mean and variance of the parameters; you can also discuss the probability that your parameter is larger or smaller than a certain value.\\
-
-\textsf{\textbf{TIP!}} These graphs are of course especially relevant and present key results, but they can also serve as tools to detect problems or build additional confidence in your results. First, the prior and the posterior distributions should not be excessively different. Second, the posterior distributions should be close to normal, or at least not display a shape that is clearly non-normal. Third, the green mode (calculated from the numerical optimization of the posterior kernel) should not be too far away from the mode of the posterior distribution. If not, it is advisable to undertake a greater number of Metropolis-Hastings simulations. \\
-
-The last figure returns the smoothed estimated shocks in a useful illustration to eye-ball the plausibility of the size and frequency of the shocks. The horizontal axis, in this case, represents the number of periods in the sample. One thing to check is the fact that shocks should be centered around zero. That is indeed the case for our example. \\
-
-
-\begin{comment}
-\section{Dealing with non-stationary models}
-Many DSGE models will embody a source of non-stationarity. This adds a layer of complication to estimating these models. In Dynare, this involves stationarizing variables, as well as several new lines of instruction. Again, the example below will be as basic as possible, building on the example used above. But you can go through a more complex example in the advanced chapter on DSGE estimation (chapter \ref{ch:estadv}). \\
-
-Below, we will introduce the non-stationarity with a stochastic trend in technology diffusion. But in order to linearize and solve the DSGE model - a necessary first step before estimating it - we must have a steady state. Thus, we must write the model using stationary variables. Our task will then be to link these stationary variables to the data which, according to the model, would be non-stationary. \textsf{\textbf{NOTE!}} But whatever the data, in Dynare, \textbf{the model must be made stationary}. \\
-
-\subsection{The source of non-stationarity}
-
-
-
-\section{Specifying observable variable characteristics}
-A natural extension of our discussion above is to declare, in Dynare, which variables are observable and which are non-stationary. This can be done just after the model block. 
-
-\subsection{Declaring observable variables}
-We begin by declaring which of our model's variables are observables. In our .mod file we write\\
-\\
-\texttt{varobs P\_obs Y\_obs;}\\
-\\
-to specify that our observable variables are indeed P\_obs and Y\_obs as noted in the section above. \textsf{\textbf{NOTE!}} The number of unobserved variables (number of \texttt{var} minus number of \texttt{varobs}) must be smaller or equal to the number of shocks such that the model be estimated. If this is not the case, you should add measurement shocks to your model where you deem most appropriate. \\
-
-\subsection{Declaring trends in observable variables}
-
-Recall that we decided to work with the non-stationary observable variables in levels. Both output and prices exhibit stochastic trends. This can be seen explicitly by taking the difference of logs of output and prices to compute growth rates. In the case of output, we make use of the usual (by now!) relationship $Y_t=\widehat Y_t \cdot A_t$. Taking logs of both sides and subtracting the same equation scrolled back one period, we find:
-\[
-\Delta \ln Y_t = \Delta \ln \widehat Y_t + \gamma + \epsilon_{A,t}
-\]
-emphasizing clearly the drift term $\gamma$, whereas we know $\Delta \ln \widehat Y_t$ is stationary in steady state. \\
-
-In the case of prices, we apply the same manipulations to show that:
-\[
-\Delta \ln P_t = \Delta \ln \widehat P_t + \ln m_{t-1} - \gamma - \epsilon_{A,t}
-\]
-
-Note from the original equation of motion of $\ln m_t$ that in steady state, $\ln m_t=\ln m^*$, so that the drift terms in the above equation are $\ln m^* - \gamma$.\footnote{This can also be see from substituting for $\ln m_{t-1}$ in the above equation with the equation of motion of $\ln m_t$ to yield: $\Delta \ln P_t = \Delta \ln \widehat P_t + \ln m^* + \rho(\ln m_{t-2}-\ln m^*) + \epsilon_{M,t} - \gamma - \epsilon_{A,t}$ where all terms on the right hand side are constant, except for $\ln m^*$ and $\gamma$.} \\
-
-In Dynare, any trends, whether deterministic or stochastic (the drift term) must be declared up front. In the case of our example, we therefore write (in a somewhat cumbersome manner)\\
-\\
-\texttt{observation\_trends;\\
-P\_obs (log(exp(gam)/mst));\\
-Y\_obs (gam);\\
-end;}\\
-
-In general, the command \texttt{observation\_trends} specifies linear trends as a function of model parameters for the observed variables in the model.\\
-
-\subsection{Declaring unit roots, if any, in observable variables}
-
-And finally, since P\_obs and Y\_obs inherit the unit root characteristics of their driving variables, technology and money, we must tell Dynare to use a diffuse prior (infinite variance) for their initialization in the Kalman filter. Note that for stationary variables, the unconditional covariance matrix of these variables is used for initialization. The algorithm to compute a true diffuse prior is taken from \citet{DurbinKoopman2001}. To give these instructions to Dynare, we write in the .mod\\
-\\
-\texttt{unit\_root\_vars = \{'P\_obs'; 'Y\_obs' \};}\\
-\\
-\textsf{\textbf{NOTE!}} You don't need to declare unit roots for any non-stationary model. Unit roots are only related to stochastic trends. You don't need to use a diffuse initial condition in the case of a deterministic trend, since the variance is finite.\\
-
-\section{SS}
-
-UNIT ROOT SS: The only difference with regards to model estimation is the non-stationary nature of our observable variables. In this case, the declaration of the steady state must be slightly amended in Dynare. (** fill in with the new functionality from Dynare version 4, currently the answer is to write a separate Matlab file which solves the steady state for the stationary variables and returns dummies for the non stationary variables). 
-
-
-
- \end{comment}
-
- 
-
-
-
-
-
-

doc/userguide/ch-estbeh.tex

-\chapter{Estimating DSGE models - Behind the scenes of Dynare} \label{ch:estbeh}
-
-This chapter focuses on the theory of Bayesian estimation. It begins by motivating Bayesian estimation by suggesting some arguments in favor of it as opposed to other forms of model estimation. It then attempts to shed some light on what goes on in Dynare's machinery when it estimates DSGE models. To do so, this section surveys the methodologies adopted for Bayesian estimation, including defining what are prior and posterior distributions, using the Kalman filter to find the likelihood function, estimating the posterior function thanks to the Metropolis-Hastings algorithm, and comparing models based on posterior distributions.
-
-
-\section{Advantages of Bayesian estimation}
-
-Bayesian estimation is becoming increasingly popular in the field of macro-economics. Recent papers have attracted significant attention; some of these include:  \citet{Schorfheide2000} which uses Bayesian methods to compare the fit of two competing DSGE models of consumption, \citet{LubikSchorfheide2003} which investigates whether central banks in small open economies respond to exchange rate movements, \citet{SmetsWouters2003} which applies Bayesian estimation techniques to a model of the Eurozone, \citet{Ireland2004} which emphasizes instead maximum likelihood estimation, \citet{VillaverdeRubioRamirez2004} which reviews the econometric properties of Bayesian estimators and compare estimation results with maximum likelihood and BVAR methodologies, \citet{LubikSchorfheide2005} which applies Bayesian estimation methods to an open macro model focussing on issues of misspecification and identification, and finally \citet{RabanalRubioRamirez2005} which compares the fit, based on posterior distributions, of four competing specifications of New Keynesian monetary models with nominal rigidities.\\
-
-There are a multitude of advantages of using Bayesian methods to estimate a model, but five of these stand out as particularly important and general enough to mention here.\\
-
-First, Bayesian estimation fits the complete, solved DSGE model, as opposed to GMM estimation which is based on particular equilibrium relationships such as the Euler equation in consumption. Likewise, estimation in the Bayesian case is based on the likelihood generated by the DSGE system, rather than the more indirect discrepancy between the implied DSGE and VAR impulse response functions. Of course, if your model is entirely mis-specified, estimating it using Bayesian techniques could be a disadvantage.\\
-
-Second, Bayesian techniques allow the consideration of priors which work as weights in the estimation process so that the posterior distribution avoids peaking at strange points where the likelihood peaks. Indeed, due to the stylized and often misspecified nature of DSGE models, the likelihood often peaks in regions of the parameter space that are contradictory with common observations, leading to the ``dilemma of absurd parameter estimates''.\\
-
-Third, the inclusion of priors also helps identifying parameters. Unfortunately, when estimating a model, the problem of identification often arises. It can be summarized by different values of structural parameters leading to the same joint distribution for observables. More technically, the problem arises when the posterior distribution is flat over a subspace of parameter values. But the weighting of the likelihood with prior densities often leads to adding just enough curvature in the posterior distribution to facilitate numerical maximization.\\
-
-Fourth, Bayesian estimation explicitly addresses model misspecification by including shocks, which can be interpreted as observation errors, in the structural equations.\\
-
-Sixth, Bayesian estimation naturally leads to the comparison of models based on fit. Indeed, the posterior distribution corresponding to competing models can easily be used to determine which model best fits the data. This procedure, as other topics mentioned above, is discussed more technically in the subsection below. 
-
-
-\section{The basic mechanics of Bayesian estimation}
-This and the following subsections are based in great part on work by, and discussions with, StŽphane Adjemian, a member of the Dynare development team. Some of this work, although summarized in presentation format, is available in the ``Events'' page of the \href{http://www.dynare.org/events/workshop-on-learning-and-monetary-policy}{Dynare website}. Other helpful material includes \citet{AnSchorfheide2006}, which includes a clear and quite complete introduction to Bayesian estimation, illustrated by the application of a simple DSGE model. Also, the appendix of \citet{Schorfheide2000} contains details as to the exact methodology and possible difficulties encountered in Bayesian estimation. You may also want to take a glance at \citet{Hamilton1994}, chapter 12, which provides a very clear, although somewhat outdated, introduction to the basic mechanics of Bayesian estimation. Finally, the websites of \href{http://www.econ.upenn.edu/~schorf/}{Frank Schorfheide} and \href{http://www.econ.upenn.edu/~jesusfv/index.html}{Jesus Fernandez-Villaverde} contain a wide variety of very helpful material, from example files to lecture notes to related papers. Finally, remember to also check the \href{http://www.dynare.org/documentation-and-support/examples}{open online examples} of the Dynare website for examples of .mod files touching on Bayesian estimation. \\
-
-At its most basic level, Bayesian estimation is a bridge between calibration and maximum likelihood. The tradition of calibrating models is inherited through the specification of priors. And the maximum likelihood approach enters through the estimation process based on confronting the model with data. Together, priors can be seen as weights on the likelihood function in order to give more importance to certain areas of the parameter subspace. More technically, these two building blocks - priors and likelihood functions - are tied together by Bayes' rule. Let's see how. \\
-
-First, priors are described by a density function of the form  
-\[ p(\boldsymbol\theta_{\mathcal A}|\mathcal A) \] where $\mathcal A$ stands for a specific model, $\boldsymbol\theta_{\mathcal A}$ represents the parameters of model $\mathcal A$, $p(\bullet)$ stands for a probability density function (pdf) such as a normal, gamma, shifted gamma, inverse gamma, beta, generalized beta, or uniform function. \\
-
-Second, the likelihood function describes the density of the observed
-        data, given the model and its parameters:
-        \[
-            {\mathcal L}(\boldsymbol\theta_{\mathcal A}|\mathbf Y_T,\mathcal A) \equiv p(\mathbf Y_T | \boldsymbol\theta_{\mathcal A}, \mathcal A)
-        \]
-        where $\mathbf Y_T$ are the observations until period
-        $T$, and where in our case the likelihood is recursive and can be written as:
-        \[
-            p(\mathbf Y_T | \boldsymbol\theta_{\mathcal A}, \mathcal A) =
-            p(y_0|\boldsymbol\theta_{\mathcal A},\mathcal A)\prod^T_{t=1}p(y_t | \mathbf Y_{t-1},\boldsymbol\theta_{\mathcal A}, \mathcal A)
-        \]
-
-We now take a step back. Generally speaking, we have a prior density $p(\boldsymbol\theta)$ on the one hand, and on the other, a likelihood $p(\mathbf Y_T | \boldsymbol\theta)$. In the end, we are interested in $p(\boldsymbol\theta | \mathbf Y_T)$, the \textbf{posterior
-        density}. Using the \textbf{Bayes theorem} twice we obtain this density of parameters
-        knowing the data. Generally, we have
-         \[p(\boldsymbol\theta | \mathbf Y_T) = \frac{p(\boldsymbol\theta ; \mathbf Y_T) }{p(\mathbf Y_T)}\]
-
-We also know that
-    \[
-        p(\mathbf Y_T |\boldsymbol\theta ) = \frac{p(\boldsymbol\theta ; \mathbf Y_T)}{p(\boldsymbol\theta)}
-        \Leftrightarrow p(\boldsymbol\theta ; \mathbf Y_T) = p(\mathbf Y_T |\boldsymbol\theta)\times
-        p(\boldsymbol\theta)
-    \]
-
-By using these identities, we can combine the \textbf{prior density} and the \textbf{likelihood function} discussed above to get the posterior density:
-        \[
-            p(\boldsymbol\theta_{\mathcal A} | \mathbf Y_T, \mathcal A) = \frac{p(\mathbf Y_T |\boldsymbol\theta_{\mathcal A}, \mathcal A)p(\boldsymbol\theta_{\mathcal A}|\mathcal A)}{p(\mathbf Y_T|\mathcal A)}
-        \]
-where $p(\mathbf Y_T|\mathcal A)$ is the \textbf{marginal density} of the data
-        conditional on the model:
-        \[
-            p(\mathbf Y_T|\mathcal A) = \int_{\Theta_{\mathcal A}} p(\boldsymbol\theta_{\mathcal A} ; \mathbf Y_T
-            |\mathcal A) d\boldsymbol\theta_{\mathcal A}
-        \]
-
-Finally, the \textbf{posterior kernel} (or un-normalized posterior density, given that the marginal density above is a constant or equal for any parameter), corresponds to the numerator of the posterior density:
-        \[
-            p(\boldsymbol\theta_{\mathcal A} | \mathbf Y_T, \mathcal A) \propto p(\mathbf Y_T |\boldsymbol\theta_{\mathcal A},
-            \mathcal A)p(\boldsymbol\theta_{\mathcal A}|\mathcal A) \equiv \mathcal K(\boldsymbol\theta_{\mathcal A} | \mathbf Y_T, \mathcal A)
-        \]
-This is the fundamental equation that will allow us to rebuild all posterior moments of interest. The trick will be to estimate the likelihood function with the help of the \textbf{Kalman filter} and then simulate the posterior kernel using a sampling-like or Monte Carlo method such as the \textbf{Metropolis-Hastings}. These topics are covered in more details below. Before moving on, though, the subsection below gives a simple example based on the above reasoning of what we mean when we say that Bayesian estimation is ``somewhere in between calibration and maximum likelihood estimation''. The example is drawn from \citet{Zellner1971}, although other similar examples can be found in \citet{Hamilton1994}, chapter 12.\\
-
-\subsection{Bayesian estimation: somewhere between calibration and maximum likelihood estimation - an example}
-
-Suppose a data generating process $y_t = \mu + \varepsilon_t$ for $t=1,...,T$, where $\varepsilon_t \sim \mathcal{N}(0,1)$ is gaussian white noise. Then, the likelihood is given by
-            \[
-                p(\mathbf{Y}_T|\mu) =
-                (2\pi)^{-\frac{T}{2}}e^{-\frac{1}{2}\sum_{t=1}^T(y_t-\mu)^2}
-            \]
-We know from the above that $\widehat{\mu}_{ML,T} = \frac{1}{T}\sum_{t=1}^T y_t \equiv
-                \overline{y}$ and that $\mathbb{V}[\widehat{\mu}_{ML,T}] = \frac{1}{T}$. \\
-                
-In addition, let our prior be a gaussian distribution with expectation
-            $\mu_0$ and variance $\sigma_{\mu}^2$. Then, the posterior density is defined, up to a constant, by:
-            \[
-                p\left(\mu|\mathbf{Y}_T\right) \propto
-                (2\pi\sigma_{\mu}^2)^{-\frac{1}{2}}e^{-\frac{1}{2}\frac{(\mu-\mu_0)^2}{\sigma_{\mu}^2}}\times(2\pi)^{-\frac{T}{2}}e^{-\frac{1}{2}\sum_{t=1}^T(y_t-\mu)^2}
-            \]
-Or equivalently, $p\left(\mu|\mathbf{Y}_T\right) \propto
-                e^{-\frac{\left(\mu-\mathbb{E}[\mu]\right)^2}{\mathbb{V}[\mu]}}$, with
-            \[
-                \mathbb{V}[\mu] = \frac{1}{\left(\frac{1}{T}\right)^{-1} +
-                \sigma_{\mu}^{-2}}
-            \]
-            and
-            \[
-                \mathbb{E}[\mu] =
-                \frac{\left(\frac{1}{T}\right)^{-1}\widehat{\mu}_{ML,T} +
-                \sigma_{\mu}^{-2}\mu_0}{\left(\frac{1}{T}\right)^{-1} +
-                \sigma_{\mu}^{-2}}
-            \]
-
-From this, we can tell that the posterior mean is a convex combination of the prior mean and the ML estimate. In particular, if $\sigma_{\mu}^2 \rightarrow \infty$ (ie, we have no prior information, so we just estimate the model) then $\mathbb{E}[\mu] \rightarrow \widehat{\mu}_{ML,T}$, the maximum likelihood estimator. But if $\sigma_{\mu}^2 \rightarrow 0$ (ie, we're sure of ourselves and we calibrate the parameter of interest, thus leaving no room for estimation) then $\mathbb{E}[\mu] \rightarrow \mu_0$, the prior mean. Most of the time, we're somewhere in the middle of these two extremes. 
-
-
-\section{DSGE models and Bayesian estimation}
-\subsection{Rewriting the solution to the DSGE model}
-Recall from chapter \ref{ch:solbeh} that any DSGE model, which is really a collection of first order and equilibrium conditions, can be written in the form $\mathbb{E}_t\left\{f(y_{t+1},y_t,y_{t-1},u_t)\right\}=0$, taking as a solution equations of the type $y_t = g(y_{t-1},u_t)$, which we call the decision rule. In more appropriate terms for what follows, we can rewrite the solution to a DSGE model as a system in the following manner:
-  \begin{eqnarray*}
-    y^*_t &=& M\bar y(\theta)+M\hat y_t+N(\theta)x_t+\eta_t\\
-\hat y_t &=& g_y(\theta)\hat y_{t-1}+g_u(\theta)u_t\\
-E(\eta_t \eta_t') &=& V(\theta)\\
-E(u_t u_t') &=& Q(\theta)
-  \end{eqnarray*}
-where $\hat y_t$ are variables in deviations from steady state, $\bar y$ is the vector of steady state values and $\theta$ the vector of deep (or structural) parameters to be estimated. Other variables are described below.\\
-
-The second equation is the familiar decision rule mentioned above. But the equation expresses a relationship among true endogenous variables that are not directly observed. Only $y^*_t$ is observable, and it is related to the true variables with an error $\eta_t$. Furthermore, it may have a trend, which is captured with $N(\theta)x_t$ to allow for the most general case in which the trend depends on the deep parameters. The first and second equations above therefore naturally make up a system of measurement and transition or state equations, respectively, as is typical for a Kalman filter (you guessed it, it's not a coincidence!). \\
-
-\subsection{Estimating the likelihood function of the DSGE model}
-The next logical step is to estimate the likelihood of the DSGE solution system mentioned above. The first apparent problem, though, is that the equations are non linear in the deep parameters. Yet, they are linear in the endogenous and exogenous variables so that the likelihood may be evaluated with a linear prediction error algorithm like the Kalman filter. This is exactly what Dynare does. As a reminder, here's what the Kalman filter recursion does.\\
-
-For $t=1,\ldots,T$ and with initial values $y_1$ and $P_1$ given, the recursion follows
-\begin{eqnarray*}
-  v_t &=& y^*_t - \bar y^* - M\hat y_t - Nx_t\\
-  F_t &=& M P_t M'+V\\
-  K_t &=& g_yP_tg_y'F_t^{-1}\\
-  \hat y_{t+1} &=& g_y \hat y_t+K_tv_t\\
-  P_{t+1} &=& g_y P_t (g_y-K_tM)'+g_uQg_u'
-\end{eqnarray*}
-For more details on the Kalman filter, see \citet{Hamilton1994}, chapter 13. \\
-
-From the Kalman filter recursion, it is possible to derive the \textbf{log-likelihood} given by
-\[
-\ln \mathcal{L}\left(\boldsymbol\theta|\mathbf Y^*_T\right) = -\frac{Tk}{2}\ln(2\pi)-\frac{1}{2}\sum_{t=1}^T|F_t|-\frac{1}{2}v_t'F_t^{-1}v_t  
-\]
-where the vector $\boldsymbol\theta$ contains the parameters we have to estimate: $\theta$, $V(\theta)$ and $Q(\theta)$ and where $Y^*_T$ expresses the set of observable endogenous variables $y^*_t$ found in the measurement equation. \\
-
-The log-likelihood above gets us one step closer to our goal of finding the posterior distribution of our parameters. Indeed, the \textbf{log posterior kernel} can be expressed as 
-\[
-\ln \mathcal{K}(\boldsymbol\theta|\mathbf Y^*_T) = \ln \mathcal{L}\left(\boldsymbol\theta|\mathbf Y^*_T\right) + \ln p(\boldsymbol\theta)
-\]
-where the first term on the right hand side is now known after carrying out the Kalman filter recursion. The second, recall, are the priors, and are also known. \\
-
-\subsection{Finding the mode of the posterior distribution}
-Next, to find the mode of the posterior distribution - a key parameter and an important output of Dynare - we simply maximize the above log posterior kernel with respect to $\theta$. This is done in Dynare using numerical methods. Recall that the likelihood function is not Gaussian with respect to $\theta$ but to functions of $\theta$ as they appear in the state equation. Thus, this maximization problem is not completely straightforward, but fortunately doable with modern computers. \\
-
-\subsection{Estimating the posterior distribution}
-Finally, we are now in a position to find the posterior distribution of our parameters. The distribution will be given by the kernel equation above, but again, it is a nonlinear and complicated function of the deep parameters $\theta$. Thus, we cannot obtain an explicit form for it. We resort, instead, to sampling-like methods, of which the Metropolis-Hastings has been retained in the literature as particularly efficient. This is indeed the method adopted by Dynare.\\
-
-The general idea of the Metropolis-Hastings algorithm is to simulate the posterior distribution. It is a ``rejection sampling algorithm'' used to generate a sequence of samples (also known as a ``Markov Chain'' for reasons that will become apparent later) from a distribution that is unknown at the outset. Remember that all we have is the posterior mode; we are instead more often interested in the mean and variance of the estimators of $\theta$. To do so, the algorithm builds on the fact that under general conditions the distribution of the deep parameters will be asymptotically normal. The algorithm, in the words of An and Shorfheide, ``constructs a Gaussian approximation around the posterior mode and uses a scaled version of the asymptotic covariance matrix as the covariance matrix for the proposal distribution. This allows for an efficient exploration of the posterior distribution at least in the neighborhood of the mode'' (\citet{AnSchorfheide2006}, p. 18). More precisely, the \textbf{Metropolis-Hastings algorithm implements the following steps}: 
-\begin{enumerate}
-        \item Choose a starting point $\boldsymbol\theta^\circ$, where this is typically the posterior mode, and run a loop over
-        2-3-4. 
-        \item Draw a \emph{proposal} $\boldsymbol\theta^*$ from a \emph{jumping} distribution
-        \[
-            J(\boldsymbol\theta^*|\boldsymbol\theta^{t-1}) =
-            \mathcal N(\boldsymbol\theta^{t-1},c\Sigma_{m})
-        \]
-        where $\Sigma_{m}$ is the inverse of the Hessian computed at the posterior mode.
-        \item Compute the acceptance ratio
-        \[
-            r = \frac{p(\boldsymbol\theta^*|\mathbf Y_T)}{p(\boldsymbol\theta^{t-1}|\mathbf
-            Y_T)} = \frac{\mathcal K(\boldsymbol\theta^*|\mathbf Y_T)}{\mathcal K(\boldsymbol\theta^{t-1}|\mathbf
-            Y_T)}
-        \]
-        \item Finally accept or discard the proposal $\boldsymbol\theta^*$  according to the following rule, and update, if necessary, the jumping distribution:
-        \[
-            \boldsymbol\theta^t = \left\{
-            \begin{array}{ll}
-                \boldsymbol\theta^* & \mbox{ with probability $\min(r,1)$}\\
-                \boldsymbol\theta^{t-1} & \mbox{ otherwise.}
-            \end{array}\right.
-        \]
-    \end{enumerate}
-
-Figure \ref{fig:MH} tries to clarify the above. In step 1, choose a candidate paramter, $\theta^*$ from a Normal distribution, whose mean has been set to $\theta^{t-1}$ (this will become clear in just a moment). In step 2, compute the value of the posterior kernel for that candidate parameter, and compare it to the value of the kernel from the mean of the drawing distribution. In step 3, decide whether or not to hold on to your candidate parameter. If the acceptance ratio is greater than one, then definitely keep your candidate. Otherwise, go back to the candidate of last period (this is true in very coarse terms, notice that in fact you would keep your candidate only with a probability less than one). Then, do two things. Update the mean of your drawing distribution, and note the value of the parameter your retain. After having repeated these steps often enough, in the final step, build a histogram of those retained values. Of course, the point is for each ``bucket'' of the histogram to shrink to zero. This ``smoothed histogram'' will eventually be the posterior distribution after sufficient iterations of the above steps.\\
-\begin{figure} \label{fig:MH}
-\begin{center} 
-\includegraphics[width=1.0\textwidth]{P_MH2} 
-\end{center} 
-\caption[Illustration of the Metropolis-Hastings algorithm]{The above sketches the Metropolis-Hastings algorithm, used to build the posterior distribution function. Imagine repeating these steps a large number of times, and smoothing the ``histogram'' such that each ``bucket'' has infinitely small width.} 
-\end{figure}
-
-But why have such a complicated acceptance rule? The point is to be able to visit the entire domain of the posterior distribution. We should not be too quick to simply throw out the candidate giving a lower value of the posterior kernel, just in case using that candidate for the mean of the drawing distribution allows us to to leave a local maximum and travel towards the global maximum. Metaphorically, the idea is to allow the search to turn away from taking a small step up, and instead take a few small steps down in the hope of being able to take a big step up in the near future. Of course, an important parameter in this searching procedure is the variance of the jumping distribution and in particular the \textbf{scale factor}. If the scale factor is too small, the \textbf{acceptance rate} (the fraction of candidate parameters that are accepted in a window of time) will be too high and the Markov Chain of candidate parameters will ``mix slowly'', meaning that the distribution will take a long time to converge to the posterior distribution since the chain is likely to get ``stuck'' around a local maximum. On the other hand, if the scale factor is too large, the acceptance rate will be very low (as the candidates are likely to land in regions of low probability density) and the chain will spend too much time in the tails of the posterior distribution. \\
-
-While these steps are mathematically clear, at least to a machine needing to undertake the above calculations, several practical questions arise when carrying out Bayesian estimation. These include: How should we choose the scale factor $c$ (variance of the jumping
-        distribution)? What is a satisfactory acceptance rate? How many draws are ideal? How is convergence of the Metropolis-Hastings iterations assessed? These are all important questions that will come up in your usage of Dynare. They are addressed as clearly as possible in section \ref{sec:estimate} of Chapter \ref{ch:estbase}. \\
-
-\section{Comparing models using posterior distributions}
-As mentioned earlier, while touting the advantages of Bayesian estimation, the posterior distribution offers a particularly natural method of comparing models. Let's look at an illustration. \\
-
-Suppose we have a prior distribution over two competing models: $p(\mathcal{A})$ and $p(\mathcal{B})$. Using Bayes' rule, we can compute the posterior
-            distribution over models, where $\mathcal{I}=\mathcal{A},\mathcal{B}$ 
-        \[
-            p(\mathcal{I}|\mathbf Y_T) = \frac{p(\mathcal{I})p(\mathbf Y_T|\mathcal{I})}
-            {\sum_{\mathcal{I}=\mathcal{A},\mathcal{B}}p(\mathcal{I})p(\mathbf Y_T|\mathcal{I})}
-        \]
-where this formula may easily be generalized to a collection of $N$ models.
-
-Then, the comparison of the two models is done very naturally through the ratio of the posterior model distributions. We call this the \textbf{posterior odds ratio}:
-        \[
-            \frac{p(\mathcal{A}|\mathbf Y_T)}{p(\mathcal{B}|\mathbf
-            Y_T)} = \frac{p(\mathcal{A})}{p(\mathcal{B})}
-            \frac{p(\mathbf Y_T|\mathcal{A})}{p(\mathbf Y_T|\mathcal{B})}
-        \]
-
-The only complication is finding the magrinal density of the data conditional on the model, $p(\mathbf Y_T|\mathcal{I})$, which is also the denominator of the posterior density $p(\boldsymbol\theta | \mathbf Y_T)$ discussed earlier. This requires some detailed explanations of their own. \\
-
-For each model $\mathcal{I}=\mathcal{A},\mathcal{B}$ we can evaluate, at least theoretically, the marginal
-        density of the data conditional on the model by integrating out the deep parameters $\boldsymbol\theta_{\mathcal{I}}$ from the posterior kernel:
-        \[
-            p(\mathbf Y_T|\mathcal{I}) = \int_{\Theta_{\mathcal{I}}} p(\boldsymbol\theta_{\mathcal{I}}; \mathbf Y_T
-            |\boldsymbol\theta_{\mathcal{I}},\mathcal{I}) d\boldsymbol\theta_{\mathcal{I}} = \int_{\Theta_{\mathcal{I}}} p(\boldsymbol\theta_{\mathcal{I}}|\mathcal{I})\times p(\mathbf Y_T
-            |\boldsymbol\theta_{\mathcal{I}},\mathcal{I}) d\boldsymbol\theta_{\mathcal{I}}
-        \]
-Note that the expression inside the integral sign is exactly the posterior kernel. To remind you of this, you may want to glance back at the first subsection above, specifying the basic mechanics of Bayesian estimation.\\
-        
-        To obtain the marginal density of the data conditional on the model, there are two options. The first is to assume a functional form of the posterior kernel that we can integrate. The most straightforward and the best approximation, especially for large samples, is the Gaussian (called a \textbf{Laplace approximation}). In this case, we would have the following estimator:
-\[
-\widehat{p}(\mathbf Y_T|\mathcal I) = (2\pi)^{\frac{k}{2}}|\Sigma_{\boldsymbol\theta^m_{\mathcal I}}|^{\frac{1}{2}}p(\boldsymbol\theta_{\mathcal I}^m|\mathbf
-Y_T,\mathcal I)p(\boldsymbol\theta_{\mathcal I}^m|\mathcal I)
-\]
-where $\boldsymbol\theta_{\mathcal I}^m$ is the posterior mode. The advantage of this technique is its computational efficiency: time consuming Metropolis-Hastings iterations are not necessary, only the numerically calculated posterior mode is required. \\
-
-The second option is instead to use information from the Metropolis-Hastings runs and is typically referred to as the \textbf{Harmonic Mean Estimator}. The idea is to simulate the marginal density of interest and to simply take an average of these simulated values. To start, note that
-\[
-p(\mathbf{Y}_T|\mathcal I)=\mathbb{E}\left[\frac{f(\boldsymbol\theta_{\mathcal I})}{p(\boldsymbol\theta_{\mathcal I}|\mathcal I)
-            p(\mathbf{Y}_T|\boldsymbol\theta_{\mathcal I},\mathcal I)}\biggl|\boldsymbol\theta_{\mathcal I},\mathcal I\biggr.\right]^{-1}
-\]
-where $f$ is a probability density function, since 
-\[
-                \mathbb{E}\left[\frac{f(\boldsymbol\theta_{\mathcal I})}{p(\boldsymbol\theta_{\mathcal I}|\mathcal I)
-                p(\mathbf{Y}_T|\boldsymbol\theta_{\mathcal I},\mathcal I)}\biggl|\boldsymbol\theta_{\mathcal I},\mathcal I\biggr.\right]
-                =
-                \frac{\int_{\Theta_{\mathcal I}}f(\boldsymbol\theta)d\boldsymbol\theta}{\int_{\Theta_{\mathcal I}}p(\boldsymbol\theta_{\mathcal I}|I)
-                p(\mathbf{Y}_T|\boldsymbol\theta_{\mathcal I},\mathcal I)d\boldsymbol\theta_{\mathcal I}}
-\]
-and the numerator integrates out to one (see\citet{Geweke1999} for more details). \\
-
-This suggests the following estimator of the marginal
-        density
-        \[
-            \widehat{p}(\mathbf{Y}_T|\mathcal I)= \left[\frac{1}{B}\sum_{b=1}^B
-            \frac{f(\boldsymbol\theta_{\mathcal I}^{(b)})}{p(\boldsymbol\theta_{\mathcal I}^{(b)}|\mathcal I)
-            p(\mathbf{Y}_T|\boldsymbol\theta_{\mathcal I}^{(b)},\mathcal I)}\right]^{-1}
-        \]
-where each drawn vector $\boldsymbol\theta_{\mathcal I}^{(b)}$ comes from the
-        Metropolis-Hastings iterations and where the probability density function $f$ can be viewed as a weights on the posterior kernel in order to downplay the importance of extreme values of $\boldsymbol\theta$. \citet{Geweke1999} suggests to use a truncated Gaussian function, leading to what is typically referred to as the \textbf{Modified Harmonic Mean Estimator}. \\    
-            
-            

doc/userguide/ch-inst.tex

-\chapter{Installing Dynare} \label{ch:inst}
-
-\section{Dynare versions}
-
-Dynare runs on both \href{http://www.mathworks.com/products/matlab/}{\textbf{MATLAB}} and \href{http://www.octave.org}{\textbf{GNU Octave}}.
-
-There used to be versions of Dynare for \textbf{Scilab} and \textbf{Gauss}. Development of the Scilab version stopped after Dynare version 3.02 and that for Gauss after Dynare version 1.2.
-
-This User Guide will exclusively \textbf{focus on Dynare version 4.0 and later}.
-
-You may also be interested by another program, \textbf{Dynare++}, which is a standalone C++ program specialized in computing k-order approximations of dynamic stochastic general equilibrium models. Note that Dynare++ is distributed along with Dynare since version 4.1. See the \href{http://www.dynare.org/documentation-and-support/dynarepp}{Dynare++ webpage} for more information. 
-
-\section{System requirements}
-Dynare can run on Microsoft Windows, as well as Unix-like operating systems, in particular GNU/Linux and Mac OS X. If you have questions about the support of a particular platform, please ask your question on \href{http://www.dynare.org/phpBB3}{\textbf{Dynare forums}}.
-
-To run Dynare, it is recommended to allocate at least 256MB of RAM to the platform running Dynare, although 512MB is preferred. Depending on the type of computations required, like the very processor intensive Metropolis Hastings algorithm, you may need up to 1GB of RAM to obtain acceptable computational times.
-
-\section{Installing Dynare}
-
-Please refer to the section entitled ``Installation and configuration'' in the \href{http://www.dynare.org/documentation-and-support/manual}{Dynare reference manual}.
-
-\section{MATLAB particularities}
-
-A question often comes up: what special MATLAB toolboxes are necessary to run Dynare? In fact, no additional toolbox is necessary for running most of Dynare, except maybe for optimal simple rules (see chapter \ref{ch:ramsey}), but even then remedies exist (see the \href{http://www.dynare.org/phpBB3}{Dynare forums} for discussions on this, or to ask your particular question). But if you do have the `optimization toolbox' installed, you will have additional options for solving for the steady state (solve\_algo option) and for searching for the posterior mode (mode\_compute option), both of which are defined later. 

doc/userguide/ch-intro.tex

-\chapter{Introduction} \label{ch:intro}
-
-Welcome to Dynare! \\
-
-\section{About this Guide - approach and structure}
-This User Guide \textbf{aims to help you master Dynare}'s main functionalities, from getting started to implementing advanced features. To do so, this Guide is structured around examples and offers practical advice. To root this understanding more deeply, though, this Guide also gives some background on Dynare's algorithms, methodologies and underlying theory. Thus, a secondary function of this Guide is to \textbf{serve as a basic primer} on DSGE model solving and Bayesian estimation. \\
-
-This Guide will focus on the most common or useful features of the program, thus emphasizing \textbf{depth over breadth}. The idea is to get you to use 90\% of the program well and then tell you where else to look if you're interested in fine tuning or advanced customization.\\
-
-This Guide is written mainly for an \textbf{advanced economist} - like a professor, graduate student or central banker - needing a powerful and flexible program to support and facilitate his or her research activities in a variety of fields. The sophisticated computer programmer, on the one hand, or the specialist of computational economics, on the other, may not find this Guide sufficiently detailed. \\
-
-We recognize that the ``advanced economist'' may be either a beginning or intermediate user of Dynare. This Guide is written to accommodate both. If you're \textbf{new to Dynare}, we recommend starting with chapters \ref{ch:solbase} and \ref{ch:estbase}, which introduce the program's basic features to solve (including running impulse response functions) and estimate DSGE models, respectively. To do so, these chapters lead you through a complete hands-on example, which we recommend following from A to Z, in order to ``\textbf{learn by doing}''. Once you have read these two chapters, you will know the crux of Dynare's functionality and (hopefully!) feel comfortable using Dynare for your own work. At that point, though, you will probably find yourself coming back to the User Guide to skim over some of the content in the advanced chapters to iron out details and potential complications you may run into.\\
-
-If you're instead an \textbf{intermediate user} of Dynare, you will most likely find the advanced chapters, \ref{ch:soladv} and \ref{ch:estadv}, more appropriate. These chapters cover more advanced features of Dynare and more complicated usage scenarios. The presumption is that you would skip around these chapters to focus on the topics most applicable to your needs and curiosity. Examples are therefore more concise and specific to each feature; these chapters read a bit more like a reference manual.\\
-
-We also recognize that you probably have had repeated if not active exposure to programming and are likely to have a strong economic background. Thus, a black box solution to your needs is inadequate. To hopefully address this issue, the User Guide goes into some depth in covering the \textbf{theoretical underpinnings and methodologies that Dynare follows} to solve and estimate DSGE models. These are available in the ``behind the scenes of Dynare'' chapters \ref{ch:solbeh} and \ref{ch:estbeh}. These chapters can also serve as a \textbf{basic primer} if you are new to the practice of DSGE model solving and Bayesian estimation. \\
-
-Finally, besides breaking up content into short chapters, we've introduced two different \textbf{markers} throughout the Guide to help streamline your reading.
-\begin{itemize}
-\item \textbf{\textsf{TIP!}} introduces advice to help you work more efficiently with Dynare or solve common problems.  
-\item \textbf{\textsf{NOTE!}} is used to draw your attention to particularly important information you should keep in mind when using Dynare. 
-\end{itemize} 
-
-
-\section{What is Dynare?}
-Before we dive into the thick of the ``trees'', let's have a look at the ``forest'' from the top \ldots just what is Dynare? \\
-
-\textbf{Dynare is a powerful and highly customizable engine, with an intuitive front-end interface, to solve, simulate and estimate DSGE models}. \\
-
-In slightly less flowery words, it is a pre-processor and a collection of Matlab routines that has the great advantages of reading DSGE model equations written almost as in an academic paper. This not only facilitates the inputting of a model, but also enables you to easily share your code as it is straightforward to read by anyone.\\
-\begin{figure} \label{fig:dyn}
-\begin{center} 
-\includegraphics[width=1.0\textwidth]{P_DynareStruct2} 
-\end{center} 
-\caption[Dynare, a bird's eyeview]{The .mod file being read by the Dynare pre-processor, which then calls the relevant Matlab routines to carry out the desired operations and display the results.} 
-\end{figure}
-
-Figure \ref{fig:dyn} gives you an overview of the way Dynare works. Basically, the model and its related attributes, like a shock structure for instance, is written equation by equation in an editor of your choice. The resulting file will be called the .mod file. That file is then called from Matlab. This initiates the Dynare pre-processor which translates the .mod file into a suitable input for the Matlab routines (more precisely, it creates intermediary Matlab or C files which are then used by Matlab code) used to either solve or estimate the model. Finally, results are presented in Matlab. Some more details on the internal files generated by Dynare is given in section \ref{sec:dynfiles} in chapter \ref{ch:soladv}. \\
-
-Each of these steps will become clear as you read through the User Guide, but for now it may be helpful to summarize \textbf{what Dynare is able to do}:
-\begin{itemize}
-\item compute the steady state of a model
-\item compute the solution of deterministic models
-\item compute the first and second order approximation to solutions of stochastic models
-\item estimate parameters of DSGE models using either a maximum likelihood or a Bayesian approach
-\item compute optimal policies in linear-quadratic models
-\end{itemize}
-
-
-\section{Additional sources of help}
-While this User Guide tries to be as complete and thorough as possible, you will certainly want to browse other material for help, as you learn about new features, struggle with adapting examples to your own work, and yearn to ask that one question whose answer seems to exist no-where. At your disposal, you have the following additional sources of help:
-\begin{itemize}
-\item \href{http://www.dynare.org/documentation-and-support/manual}{\textbf{Reference Manual}}: this manual covers all Dynare commands, giving a clear definition and explanation of usage for each. The User Guide will often introduce you to a command in a rather loose manner (mainly through examples); so reading corresponding command descriptions in the Reference Manual is a good idea to cover all relevant details. 
-\item \href{http://www.dynare.org/documentation-and-support/examples}{\textbf{Official online examples}}: the Dynare website includes other examples - usually well documented - of .mod files covering models and methodologies introduced in recent papers. 
-\item \href{http://www.dynare.org/phpBB3}{\textbf{Dynare forums}}: this lively online discussion forum allows you to ask your questions openly and read threads from others who might have run into similar difficulties. 
-\item \href{http://www.dynare.org/documentation-and-support/faq}{\textbf{Frequently Asked Questions}} (FAQ): this section of the Dynare site emphasizes a few of the most popular questions in the forums. 
-\item \href{http://www.dsge.net}{\textbf{DSGE.net}}: this website, run my members of the Dynare team, is a resource for all scholars working in the field of DSGE modeling. Besides allowing you to stay up to date with the most recent papers and possibly make new contacts, it conveniently lists conferences, workshops and seminars that may be of interest. 
-\end{itemize}
-
-\section{Nomenclature}
-To end this introduction and avoid confusion in what follows, it is worthwhile to agree on a few \textbf{definitions of terms}. Many of these are shared with the Reference Manual. 
-\begin{itemize}
-\item \textbf{Integer} indicates an integer number.
-\item \textbf{Double} indicates a double precision number. The following syntaxes are valid: 1.1e3, 1.1E3, 1.1E-3, 1.1d3, 1.1D3
-\item \textbf{Expression} indicates a mathematical expression valid in the underlying language (e.g. Matlab).
-\item \textbf{Variable name} indicates a variable name. \textbf{\textsf{NOTE!}} These must start with an alphabetical character and can only contain other alphabetical characters and digits, as well as underscores (\_). All other characters, including accents, and \textbf{spaces}, are forbidden. 
-\item \textbf{Parameter name} indicates a parameter name which must follow the same naming conventions as above. 
-\item \textbf{Filename} indicates a file name valid in your operating system. Note that Matlab requires that names of files or functions start with alphabetical characters; this concerns your Dynare .mod files.
-\item \textbf{Command} is an instruction to Dynare or other program when specified. 
-\item \textbf{Options} or optional arguments for a command are listed in square brackets \mbox{[ ]} unless otherwise noted. If, for instance, the option must be specified in parenthesis in Dynare, it will show up in the Guide as [(option)].
-\item \textbf{\texttt{Typewritten text}} indicates text as it should appear in Dynare code.
-\end{itemize}
-
-\section{v4, what's new and backward compatibility}
-The current version of Dynare - for which this guide is written - is version 4. With respect to version 3, this new version introduces several important features, as well as improvements, optimizations of routines and bug fixes. The major new features are the following: 
-\begin{itemize}
-\item Analytical derivatives are now used everywhere (for instance, in the Newton algorithm for deterministic models and in the linearizations necessary to solve stochastic models). This increases computational speed significantly. The drawback is that Dynare can now handle only a limited set of functions, although in nearly all economic applications this should not be a constraint. 
-\item Variables and parameters are now kept in the order in which they are declared whenever displayed and when used internally by Dynare. Recall that in version 3, variables and parameters where at times in their order of declaration and at times in alphabetical order. \textbf{\textsf{NOTE!}} This may cause some problems of backward compatibility if you wrote programs to run off Dynare v3 output.
-\item The names of many internal variables and the organization of output variables has changed. These are enumerated in details in the relevant chapters. The names of the files internally generated by Dynare have also changed. (** more on this when explaining internal file structure - TBD)
-\item The syntax for the external steady state file has changed. This is covered in more details in chapter \ref{ch:solbase}, in section \ref{sec:findsteady}. \textbf{\textsf{NOTE!}} You will unfortunately have to slightly amend any old steady state files you may have written. 
-\item Speed. Several large-scale improvements have been implemented to speed up Dynare. This should be most noticeable when solving deterministic models, but also apparent in other functionality.
-\end{itemize}
-
-
- 
-

doc/userguide/ch-ramsey.tex

-\chapter{Optimal policy under commitment} \label{ch:ramsey}
\ No newline at end of file